nbUfnbujljt!txbwmfcb!!J!lmbtTj! - Ganatleba

- 1 - 

q o r b u d a 

nbUfnbujljt!txbwmfcb!!J!lmbtTj! 

nfUpejlvsj!tbyfmn[Swbofmp!nbtxbwmfcmjtUwjt! 

avtorebi: 

zurab vaxania 

fsiqologiis institutis direqtoris moadgile, 

pedagogikis mecn. doqtori, 

maTematikis mecn. kandidati, 

ganaTlebis mecnierebaTa akademikosi. 

dodo naziriSvili 

pedagogikis mecn. kandidati, 

umaRlesi kategoriis maswavlebeli. 

saredaqcio sabWo: 

Tamar bardaveliZe, elza goCitaSvili, 

manana begiaSvili, xaTuna svianaZe 

sagamomcemlo samuSaoTa Semsrulebelni: 

zurab vaxania, naTela Sulaia, nikoloz vaxania 

© z.vaxania, 2006 III gamocema tel. 37-28-88 893-94-53-04 

 

maswavlebelma jer dawyebiTi klasebis zogadi meTodikuri 

saxelmZRvanelo unda waikiTxos. _ es wigni, iseve rogorc 

sxva meTodikuri saxelmZRvaneloebi, internetSicaa ganTavsebuli, 

misamarTze: www.uznadzeinstitute.ge → saxelmZRvaneloebi).

- 2 - 

s a r C e v i 

1. I klasis saxelmZRvaneloTa krebuli 

`saymawvilo maTematika~ 1 

2. aucilebeli TvalsaCinoeba da 

saswavlo nivTebi 4 

3. weris swavleba, xazva da xatva 7 

4. I klasis bolos misaRwevi Sedegebi 

saxelmwifo saswavlo gegmebis mixedviT 11 

5. programa, misi ZiriTadi Temebi 14 

6. ra unda icodes moswavlem saswavlo 

wlis bolos 17 

7. ZiriTadi saprogramo xazebis aRwera _ 

Sesabamisoba sax. saswavlo gegmasTan 22 

8. sagakveTilo monakveTebi da 

gakveTilebis ageba 34 

gakveTilebis gegma-konspeqtebi 40 

esaa pirveli klasis ZiriTadi saxelmZRvanelo, 

romliTac unda ixelmZRvanelos maswavlebelma 

(da ara mxolod moswavlis saxelmZRvaneloTi). 

maswavleblisTvis aucilebelia, rom Seasrulos 

aq aRwerili saklaso samuSaoebi - umetesad 

zepiri, saTanado TvalsaCinoebaze damyarebiT. 

mxolod moswavlis saxelmZRvaneloTi 

muSaoba Zalian arasakmarisia! 

saxelmZRvanelos agebuleba, zogadi 

samecniero-meTodikuri safuZvlebi da ZiriTadi 

maTematikuri sakiTxebis damuSavebis meTodika 

axalia da Seqmnilia avtorebis mier. 

amasTan, gamoyenebulia guram beriSvilisa da iaTamze 

kotetiSvilis calkeuli meTodikuri ideebi da 

t.k. Jikalkinaseuli zogierTi maTematikuri TamaSi.

- 3 - 

§ 1. I klasis saxelmZRvaneloTa krebuli 

`saymawvilo maTematika~ 

ganviTareba da ganaTleba aravis ar SeiZleba gadaeces 

an ecnobos. visac swadia maT eziaros, sakuTari 

rudunebiTa da nebisyofiT unda miaRwios. garedan is 

SeiZleba mxolod waxalisdes da marTebulad iqnes 

warmarTuli... amitom TviTmoqmedeba _ saSualebacaa da 

Tan nayofic ganaTlebisa. a. distervegi 

krebuli Sedgeba xuTi nawilisagan: 

1) meTodikuri saxelmZRvanelo maswavlebelTaTvis (zogadi 

nawili) _ saerToa I-IV klasebisaTvis; SemoklebiT: zms 

igi maswavlebelma pirvel rigSi unda waikiTxos! 

2) meTodikuri saxelmZRvanelo I klasis maswavleblisTvis; 

3) UwbmtbDjopfcjt! ! xjhoj _ mcire saxelmZRvanelo 

moswavleTaTvis, romelSic mocemulia mxolod ZiriTadi 

TvalsaCinoeba (moswavleebi am wignSi ar weren da ar xataven, 

mas inaxaven sufTad); SemoklebiT: mTw 

4)![jsjUbej!!swfvmj _ masSi moswavleebi xataven, xazaven, 

aferadeben, weren, klasSi iniSnaven da Sin asruleben mcire 

saSinao davalebas. 

5)! ebnbufcjUj! ! swfvmj _ iseTivea, rogorc ZiriTadi, 

oRond moicavs mxolod saSinao davalebebs, Tanac, damatebiT 

saSinao davalebebs. SedarebiT Zlieri moswavleebisTvis es 

rveuli Zalian saWiroa, danarCenebisTvis _ sasurvelia. masSi 

iwyeba logikis programa, amitom gansakuTrebiT kargia zogad 

gonebriv unarTa gansaviTareblad. 

damatebiTi rveuli pirveli erTi Tvis ganmavlobaSi sul 

araa saWiro, misi Semotana upriania meore Tvidan (Tumca, 

sjobs gvian, vidre arasdros). SemoklebiT: msr 

sxva rveuli moswavles pirvel klasSi ar dasWirdeba. 

msr-s iseTive naxatebiani yda aqvs, rogoric mTw-s, oRond 

naxatebi gauferadebelia. isini TviTon moswavlem unda gaaferados 

Tavisufal dros. nimuSad SeuZlia gamoiyenos mTw-s

- 4 - 

feradi yda. sazogadod, ydis suraTi Zalian mniSvnelovani 

TvalsaCinoebaa, romelic mravaljer gamoiyeneba gakveTilebis 

drosac. am suraTze SekumSulad da xatovnad gadmocemulia 

pirveli klasis maTematikis programis yvela ZiriTadi sakiTxi, 

Tanac bunebrivad Serwymuli saxiT samive xazi: ariTmetika, 

geometria da gamoyenebiTi maTematika. amis safuZveli sivrciTi 

mimarTebebia _ iseve, rogorc es iyo istoriulad. yovelive 

amis gamo moswavlem kargad unda icodes ydis suraTi. 

mTw-s ZiriTadi nawilia `dominosebri~ TvalsaCinoeba. yoveli 

ricxvi 1-dan 20-mde warmodgenilia lamazi da adviladdasamaxsovrebeli 

ganlagebis mqone groviT (patara wreebisa _ ix. ydis suraTze). 

garda amisa, Zalian TvalsaCinodaa mocemuli Sekreba-gamoklebis 

yvela ZiriTadi magaliTi. grovaTa ganlagebani zustad 

ar emTxveva namdvili dominos (kamaTlis) wreebis Sesabamis ganlagebas. 

amitom vwerT brWyalebSi sityvas `dominosebri~. 

mTw-Si mocemulia ricxvTa kibeebi, romlebsac moswavleebi 

Zalian xSirad gamoiyeneben; agreTve TvalsaCino sqema, Tu 

ramdeni aklia ricxvs ocamde. boloSi mocemulia Zalian 

mniSvnelovani geometriuli TvalsaCinoeba: yvela naswavli 

nakvTis saerTo dajgufebis Semajamebeli suraTi. gegmakonspeqtebSi 

zustadaa miTiTebuli, Tu romel sagakveTilo 

monakveTSi romeli TvalsaCinoeba unda gamoviyenoT da rogor. 

msr-Si mocemulia mxolod amocana-davalebebi, Zalian mravalferovanebi. 

isini gadanomrilia zustad gakveTilebis 

mixedviT. maTi numeracia emTxveva sagakveTilo monakveTebis 

numeracias meTod-saxelmZRvaneloSi. magaliTad, davalebebs 

nomrebiT 20.1, 20.2 da 20.3 (araumetes samisa!) asruleben me- 

20 gakveTilze klasSi. xolo davalebebi nomrebiT 20.5, 20.6 

da ase Semdeg _ miecemaT saSinao davalebad imave me-20 

gakveTilze da isini ganixileba momdevno 21-e gakveTilze. 20.4 

ki _ esaa saklaso TamaSi, romelic mocemulia mxolod meTodsaxelmZRvaneloSi 

(TamaSi yvelgan meoTxe nomriTaa). 

moswavles ar moeTxoveba da arc sWirdeba amocanebis 

nomrebis wakiTxva _ numeracia mxolod maswavleblisTvi-

- 5 - 

saa! zogi amocanis amoxsna iwyeba klasSi, magram mTavrdeba Sin, 

an piriqiT, magaliTad: 32.7. 33.2. es imas niSnavs, rom es 

nomeri klass saSinao davalebad miecema 32-e gakveTilis 

Semdeg, xolo momdevno 33-e gakveTilze igi gamoiyeneba me-2 

sagakveTilo monakveTSi. aseve, Tuki amocanis ormagi numeraciaa 

95.3. 95.7., es imas niSnavs, rom amocana gairCeva 95-e 

gakveTilze (me-3 sagakveTilo monakveTSi), magram nawilobriv. 

xolo morCenili nawili klass saSinao davalebad miecema da 

gairCeva momdevno 96-e gakveTilze (mis dasawyisSi, saSinao 

davalebaTa garCevisas). cxadia, moswavleebisaTvis amas aravi- 

Tari mniSvneloba ara aqvs da arc amis axsnaa saWiro. 

zogi nomeri sul gamotovebulia. es imas niSnavs, rom 

Sesabamis sagakveTilo monakveTs ar sWirdeba moswavlis rveuli 

(mxolod sazepiro saklaso samuSaoebia). iSviaTad ki sagakveTilo 

monakveTebis Tanmimdevroba odnav Secvlilia, magaliTad, 

jer 72.2-ia da Semdegaa 72.1. es teqnikur-poligrafiuli 

mizezebiTaa gamowveuli, did uxerxulobas ar Seqmnis. 

oRond masw.-i cota ufro yuradRebiT unda iyos. 

ZiriTadi msr gankuTvnilia saklaso samuSaoebisaTvisac da 

saSinao davalebebisaTvisac, damatebiTi _ mxolod saSinao 

davalebebisaTvis. mTw-s moswavle iyenebs klasSi masw.-is 

miTiTebiT, agreTve Sin yovelTvis, roca gauWirdeba raimes 

gonebaSi warmodgena. maswavlebelma unda auxsnas klass, rom 

Sinac daixmarion TvalsaCinoebis wigni! 

saSinao davalebis piroba dabeWdilia vardisfer fonze. 

isini moswavles vinmem unda waukiTxos Sin. masw.-ma unda 

ganumartos mSoblebs: saWiroa am davalebaTa mxolod wakiTxva, 

Tuki bavSvi ver gaigebs, wakiTxva kidev erTxel an 

ramdenjerme. mSobels moeTxoveba, rom dasvas bavSvi samecadinod 

da yuradReba miaqcios, mecadineobs Tu ara. mxolod 

Tavidan SeiZleba davexmaroT bavSvs pirobis gaazrebaSi, 

male saSualo gonebrivi SesaZleblobis bavSvi Tvi-

- 6 - 

Ton SeZlebs pirobis gaazrebas _ Tuki mas mivaCvevT damoukidebel 

fiqrs. amitom mSoblis daxmareba, gansakuTrebiT, 

daxmareba davalebis SesrulebaSi _ TviTon bavSvis- 

Tvis iqneba Zalian mavnebeli da Seaferxebs mis ganviTarebas! 

mSobeli unda miexmaros bavSvs mxolod gamonaklis 

SemTxvevebSi: avadmyofobisas an avadmyofobisa Tu gacdenis 

Semdeg, Zlieri daRlisas da sxva (ar vgulisxmobT Seferxebuli 

gonebrivi ganviTarebis mqone bavSvebs). 

ukanaskneli nawilis davalebaTa pirobebi aRaraa dabeWdili 

vardisfer fonze, Tanac, ukve msxvili asoebiTaa. es imitom, 

rom am droisaTvis moswavlem ukve unda icodes mcireodeni 

kiTxva, raTa TviTonac SeZlos pirobis wakiTxva. Tumca, Tuki 

zogierT bavSvs wakiTxva gauWirdeba, mas isev vinmem unda 

waukiTxos Sin, amiT araferi gafuWdeba. 

meTod-saxelmZRvaneloSi dawvrilebiTaa mocemuli yoveli 

sagakveTilo monakveTis ZiriTadi saklaso SekiTxvebi, maTze 

mosalodneli pasuxebi, sakiTxis damuSavebis ZiriTadi gegma, 

calkeuli aucilebeli ganmartebani. mocemulia saSinao 

davalebis amocanaTa ganxilvac. 

lbmfoebsvmj!qsphsbnb mihyveba gegma-konspeqtebs _ yovel 

gakveTils aqvs nomerica da mokle saTauric. 

§ 2. aucilebeli TvalsaCinoeba da saswavlo nivTebi 

1. yvela TemisTvis Zalian mosaxerxebelia da kargia 

nbhojuvsj!ebgb. esaa daaxloebiT 40-60-80 sm an ufro didi 

zomebis xis CarCoze daWedebuli an gadakruli Tunuqis 

furceli. masze ekroba patar-patara magnitebiani muyaos 

naWrebi (cifrebi, nakvTebi da sxva). Tuki verafriT ver 

moxerxdeba magnituri dafis Sovna, maSin SeiZleba misi Secvla 

imave zomebis mqone lursmnebiani ficriT (lursmnebze ikideba 

cifrebiani Tu nakvTebiani muyaos naWrebi). lursmnebi xSirxSirad 

unda iyos daWedebuli.

- 7 - 

gegma-konspeqtebSi Cven vwerT: `dafaze miakravs cifrs (an 

magnits; anda rgols, wres da sxva)~. Tuki klasSi araa 

magnituri dafa, maSin masw.-ma unda icodes, rom amis nacvlad 

gamoiyenos Sesabamisad an moZravi anbanis ficari, an 

lursmnebiani ficari, anda dafa-magidaze (mis Sesaxeb ix. 

qvemore) daawyos saTvleli nivTebi. 

2. tbUwmfmj! ojwUfcj: magnitur dafaze misakravi patara 

(daaxloebiT 2 sm zomis) wreebi, kvadratebi, jvrebi, 

samkuTxedebi da sxva nakvTebi _ TviTeuli 10-20 cali. garda 

amisa, daaxloebiT 30 cali erTi ferisa da amdenive meore 

feris patara kubiki an raime sxva nivTi; 50-mde erTnairi 

Cxiri, Zveli flomasteri, feradi muyao, rgolebi da sxva. 

saWiroa, rom klasSi ramdenime sxvadasxva saTvleli nivTebis 

nakrebi iyos. masw.-ma aucileblad unda cvalos xolme isini 

_ erTxel rom Cxirebs gamoiyenebs, meorejer rgolebi an kubikebi 

gamoiyenos da ase Semdeg (raTa moxdes ariTmetikuli moqmedebis 

nivTebisgan ganyeneba, moqmedeba raime nivTebs ar miejaWvos). 

gegma-konspeqtebSi Cven vwerT `nivTebi~ da vgulisxmobT, 

rom masw.-i TviTon cvlis xolme maT! 

3. didi (7-15 sm zomisa an ufro didi) lvcfcj _ 10-20 

cali. kidev saWiroa oTxkuTxa piramida da samkuTxa prizma 

(eseni SeiZleba sxvagvarebic iyos _ maTi naxazi ix. zms $ 4). 

saklaso TvalsaCinoebad gamosayenebeli kubebi da sxva 

sxeulebi xisa an plastmasisa unda iyos, sasurvelia, mTliani 

xisa (SigniT sicarielis gareSe), Tanac, wveroebi da wiboebi 

mocveTili ar unda hqondes! 

4. hfpnfusjvmj! oblwUfcj: wreebi, kvadratebi, marTkuTxedebi, 

samkuTxedebi, jvrebi, varskvlavebi, xuTkuTxedebi, eqvskuTxedebi, 

rvakuTxedebi _ feradi da TeTri muyaosi, plastmasis 

an faneris, zogi erTnairi ferisa da formis da sxvadasxva 

zomis, zogi erTnairi zomisa da formis da sxvadasxva feris, 

zogi ki zustad erTnairi. TviTeuli tipis nakvTi _ 2-dan 10-

- 8 - 

mde cali (jvrebi, varskvlavebi, xuTkuTxedebi, eqvskuTxedebi 

da rvakuTxedebi _ SedarebiT naklebi raodenobiT). amasTan, 

yvela nakvTi ar unda iyos tolgverdebiani da simetriuli, 

zogi arasimetriulic unda iyos! Zalian kargi iqneba, Tuki am 

nakvTebs dawebebuli eqneba patara magnitebi _ magnitur dafaze 

misakravad. nakvTebis saSualo sigrZe _ 7 sm. 

nakvTebis formebi ix. mTw-s boloSi. 

5. sjdywUb! ljcffcjt kedelze Camosakidebeli ori feradi 

suraTi (daaxloebiT 50-80 sm zomebisa). isini unda daixazos 

ise, rogorc mocemulia mTw-s boloSi. 

garda amisa, mcire droiT (ramdenime gakveTili) saWiroa 

sxvagvari ricxvTa kibec: romelsac aweria ara cifrebi, aramed 

`dominosebri ricxvebi~ _ rogorebicaa mTw-s boloSi. magaliTad, 

cifri 4-is adgilas unda iyos dasmuli patara wreebis `dominosebri~ 

oTxeuli (kvadratulad ganlagebuli) da ase Semdeg. 

Tuki amgvari saklaso TvalsaCinoebis momzadeba Zalian gaZneldeba, 

masw.-s advili gamosavali aqvs: Cveulebriv ricxvTa 

kibis suraTze (saklaso plakatze) cifrebs droebiT sakrobiT 

gadaawebos baraTebi _ saTanado raodenobaTa dominosebri ganlagebiT. 

Semdeg es baraTebi moixsneba da darCeba Cveulebrivi 

ricxvTa kibe. 

Cveneuli meTodika did yuradRebas uTmobs ricxvTa kibeze 

moqmedebebs. es Zalian advilia da TiTqos arafers aviTarebs. 

magram, sinamdvileSi, Zalian saWiroa: ricxvTa kibeze advilad 

sruldeba Tundac aTmoqmedebiani Sekreba-gamoklebac ki. 

moswavle sanaxevrod TamaSiT iangariSebs, magram amis mravaljer 

gakeTebis Semdeg TavisTavad daamaxsovrdeba gamoTvlebi, 

bolos ki zepirad, gonebaSic SeZlebs! da bavSvs ar SeeSindeba 

grZeli, mravalmoqmedebiani gamosaxulebisa! 

6. magnitur dafaze misakravi an np[sbwj! bocbojt! djgsfcj 

(TviTeuli ramdenime cali), niSnebi +, _, >, = da wertili · 

Cveulebriv, moZrav anbanebSi aris xolme mxolod cifrebi

- 9 - 

0-dan 9-mde. magram saWiroa agreTve 10, 11, 12, ... , 29 (TiTo 

cali). isini sagangebod unda damzaddes (yovelTvis or-oris 

SeerTeba Zalian mouxerxebelia). 

7. didi, fsUnfusjboj! tbyb{bwj, romelzec mkafiod Cans 

cifrebi da danayofebi. 

8. ebgb.nbhjeb saWiroa imisaTvis, raTa mTeli klasi 

kargad xedavdes erTi gamoZaxebuli moswavlis mier 

Sesrulebul moqmedebebs nivTebze. esaa magidaze dasadgmeli 

daxrilzedapiriani yuTi an ficari, naWergadakruli (raTa zed 

dadebuli nivTebi ar srialebdes), romelic magidis zedapiridan 

10-30 sm-is simaRleze iqneba. 

moswavleebs skolaSic da Sinac dasWirdebaT Semdegi 

saswavlo nivTebi: saTvleli nivTebi 30-35 cali (mobrtyelo 

Cxirebi, Rilebi, nuSebi, rgolebi an raime sxva, rac arc 

mZimea, Zalian arc srialebs da arc goravs); kalmistari, Savi 

da feradi fanqrebi (ori-sami ZiriTadi feris mainc); saSleli; 

saxazavi wriuli, kvadratuli da sxva xvrelebiT (nakvTebis 

dasaxazavad); Cveulebrivi santimetriani danayofebiani saxazavi 

dasWirdebaT daaxloebiT meSvide Tvidan. 

garda yovelive amisa, Zalian mniSvnelovani TvalsaCinoebaa 

TviT mTw-s ydis suraTi. am suraTze erTianadaa Sejamebuli 

pirveli klasis ZiriTadi ariTmetikuli da geometriuli 

sakiTxebi. es suraTi xSirad gamoiyeneba gakveTilebze. magram 

mas sxva mniSvnelobac aqvs: bavSvi xSirad daaTvalierebs mas 

da uneburad, Zaldautaneblad aRebeWdeba gonebaSi ricxvTa 

agebuleba da Sesabamisi geometriuli nakvTebi. Tanac, roca 

wigni daxurulia, wina mxridan moCans mxolod raodenobani 10is 

farglebSi, ukana mxares ki 11-dan 20-mde raodenobania. 

gakveTilze ganxilvisas suraTi mTlianad unda gamoCndes. 

amisaTvis saWiroa, rom wigni Suaze gaiSalos da merxze 

daidos gaSlilad, oRond ydiT zemoT. 

§ 3. weris swavleba, xazva da xatva 

sadac davalebaSi weria `daxaze~ _ igulisxmeba, rom

- 10 - 

moswavlem unda gamoiyenos saxazavi da daxazos SedarebiT 

zustad. sadac weria `daxate~ _ igulisxmeba, rom moswavlem 

xeliT, usaxazovod unda daxazos anu daxatos geometriuli 

nakvTi. Tumca, es araa mkacri moTxovnebi da moswavles 

SeuZlia gaakeTos ise, rogorc eadvileba. 

Tavdapirvelad geometriuli nakvTi unda iyos mTliani 

xativiT, swored rogorc xati, romelsac bavSvi 

xatavs _ iseve rogorc xatavs saxls, kacunasa Tu mzes. 

cxadia, pirvelklasels ar ZaluZs namdvili xazva Tu xatva. 

amitom yvelam xazos, xatos da weros ise, rogorc SeZlebs 

_ droisa da Zalebis dauxarjavad. es unda icodes 

mSobelmac da Tavad moswavlemac. 

masw.-ma moswavleebs dafaze unda aCvenos nimuSebi imisa, Tu 

rogor SeiZleba TviTmfrinavis, gemis, Citisa Tu sxva ramis 

àdvilad daxatva~: _ ai, viTom es Citia! (da daxatavs 

gamartivebul xazebs). amis Semdeg moswavleebi TandaTan 

mieCvevian gamartivebuli (sqematuri!) naxatebis Sesrulebas. es 

maT kargad Seamzadebs SemdgomSi namdvili maTematikuri 

sqemebis saswavladac. 

saswavlo wlis dasawyisSi bevri zedmeti dro ixarjeba 

imitom, rom moswavleebi ar arian miCveulni feradi fanqrebis 

xmarebas. es SesaWiri xatvis (xelovnebis) gakveTilebze unda 

mogvardes. damoukideblad maTematikisagan, xatvis masw.-s isedac 

evaleba, rom pirvelsave gakveTilebze bavSvebs aswavlos, Tu 

romelia wiTeli fanqari, mwvane Tu lurji fanqari da rogor 

unda maTi xmareba. es araa maTematikis saqme. 

asea Tu ise, or-sam kviraSi moswavleebi mieCvevian rveulSi 

muSaobas da saqmec aewyoba. magram, yvela SemTxvevaSi, masw.-ma 

unda icodes, rom mTavaria ara rveulebSi muSaoba, aramed 

saklaso muSaoba nivTebiTa da msjelobiT _ rac gegmakonspeqtebSia 

aRwerili. 

dawyebiTi klasebis masw.-Ta umravlesoba miCveulia, rom 

swavlebis Sedegi Seafasos imiT, Tu ramdenad kargad gamoiyu-

- 11 - 

reba moswavlis rveuli. maTematikis swavlebisas es dauSvebelia. 

swavlebis namdvil Sedegs gviCvenebs ara rveulebi, 

aramed is, Tu rogor msjeloben moswavleebi, rogor 

angariSoben zepirad, maTematikuri TvalsazrisiT ramdenad 

kargad xsnian amocanebs. I klasSi araa saWiro didi 

yuradRebis gamaxvileba kaligrafiaze, miT umetes, naxazebisa 

Tu naxatebis xarisxze. 

wera calke saprogramo Temaa. weris naadrevad dawyeba da 

masze didi yuradRebis gamaxvileba Zlier amuxruWebs bavSvis 

mier maTematikis Seswavlasa da sazogadod, misi azrovnebis 

ganviTarebas. bevri wera mZime tvirTad awveba bavSvs, abrkolebs 

misi maTematikuri azrovnebis ganviTarebas da meqanikurobisken 

ubiZgebs mas. bavSvisTvis mTavari xdeba Canaweris 

gakeTeba, da ara sakiTxis maTematikuri Tu logikuri arsi. 

weris process maTematikasTan araferi saerTo ara aqvs da 

STanTqavs ki eqvswliani bavSvis gonebrivi da sulieri Zalebis 

sakmaod did nawils maTematikis gakveTilebze. maTematika 

cifrebis xatvad gadagvardeba. Cveulebriv, pirvelklaseli 

bavSvisaTvis maTematika _ esaa cifrebis xatvis xeloba da 

agreTve 20-is farglebSi jamebisa da sxvaobebis dazepireba. 

amas ki maTematikasTan da sazogadod, azrovnebasTan TiTqmis 

araferi aqvs saerTo. 

Cveneuli programiT weris Tema Tavidan lamazi naWrelebis 

xatvasaviTaa, oRond, ujredebian badeSi (weris elementebi). 

TiTqmis oTxi Tvis ganmavlobaSi moswavleebi asruleben am 

savarjiSoebs, romlebsac ori daniSnuleba aqvs: weris 

Semzadeba da, rac mTavaria, azrovnebis ganviTareba. isini isea 

SerCeuli, rom mizanmimarTulad aviTareben ritmis, simetriis, 

sivrciTi harmoniis grZnobebs, rac TandaTan TvalsaCino 

logikur kanonzomierebaSi gadaizrdeba. mesame TveSi es savarjiSoebi 

ukve logikur datvirTvasac Seicavs. 

Tavidan, pirveli ori Tvis ganmavlobaSi, ar iswavleba 

araviTari Canawerebi da maTematikuri niSnebi: arc cifrebi, 

arc =, , + da _ niSnebi _ arc mxolod wakiTxviT!

- 12 - 

ariTmetikis xaziT iswavleba mxolod namdvili raodenobani da 

maTze moqmedebani, zepiri Sekreba-gamokleba. 

Semdeg moswavleebi swavloben cifrebis mxolod wakiTxvas. 

kidev Semdegi ori Tvis ganmavlobaSic moswavleebi cifrebs ar 

weren. dafasTan muSaobisas cifrebs wers masw.-i, moswavleebi 

mxolod kiTxuloben. klasSi roca saWiro iqneba, rom bavSvma 

daweros cifri an mTeli moqmedebis Canaweri, viyenebT 

magnitur dafas an moZravi anbanis cifrebsa da niSnebs. 

mexuTe Tvis dasawyisidan klasi iwyebs cifrebis weras. 

magram am drosac mTavari kvlav wera ki araa, aramed _ azrovneba. 

TiTqmis yvela weris savarjiSo _ saazrovno amocanas- 

Tanaa Serwymuli. weras daqvemdebarebul mniSvnelobas vaniWebT. 

sazogadod, saSualo pirvelklaseli anu 5-6-wliani bavSvis 

TiTebis moZraobis koordinaciis unari fiziologiurad araa 

imdenad momwifebuli, rom wvril-wvrili moZraobani sakmarisad 

zustad Seasrulos (mas Rilis an fexsacmlis Wapis Sekvrac 

ki uWirs!). amitom masw.-i araviTar SemTxvevaSi ar unda ai- 

Zulebdes bavSvs, rom maincadamainc kargad daxvewos cifrebis 

wera. es meore klasidan unda daiwyos. pirvel klasSi ki bavSvma 

bevri ar unda weros! man bevri unda ifiqros, imoqmedos 

saTvlel nivTebze, iangariSos, xazos da xatos, daakvirdes geometriul 

formebsa Tu TvalsaCino logikur kanonzomierebs. 

sakuTriv weris mxriv moswavles moeTxoveba mxolod sami 

ram: mieCvios ujredebian badeSi muSaobas, sworad ekavos kalmistari 

(fanqari) da nbsUfcvmj! njnbsUvmfcjU werdes cifrebs 

_ sufTad, mondomebiT da SeZlebisamebr lamazad. amisaTvis 

ki araa saWiro bevri wera da, miT umetes, maTematikis gakveTilebze 

weraze bevri drois daxarjva. 

msr-Si yoveli gakveTilis mexuTe nomeri davaleba _ saweri 

an weris winare savarjiSoa: ujredebian badeSi mocemuli 

nimuSebis daxazva Tu dawera: nimuSebze xazis gadatareba da 

aseve gagrZeleba striqonis bolomde. Tavidan, daaxloebiT 

pirveli aTi gakveTilis ganmavlobaSi, es savarjiSoebi klasSi 

keTdeba, raTa masw.-ma merxebs Soris CamovliT Seamowmos, ro-

- 13 - 

gor ukaviaT moswavleebs fanqari Tu kalmistari, rogor sxedan, 

kalmistris wveris rogori moZraobiT xazaven, xom ar jRabnian. 

es yvelaferi ixveweba imdenad, ramdenadac SeZlebs TviTeuli 

moswavle Zaldautaneblad: zogi metad, zogi naklebad. 

SemdgomSi amgvari savarjiSoebi (mexuTe nomrebi) klass saSinao 

davalebad eZleva. drodadro, daaxloebiT TveSi or-orjer, 

mas klasSic Seasruleben, raTa masw.-ma isev Seamowmos, moswavleebs 

rame xom ar eSlebaT, rames cudad xom ar eCvevian. 

rogorc iTqva zemore, TandaTanobiT Semodis arsebiTi siaxle: 

kanonzomiereba. saWiro iqneba yuradReba mieqces nimuSTa 

kanonzomier ganmeorebas ise, rogorc rveulSia mocemuli: Ti- 

To ujredis gamotovebiT, xazTa kanonzomieri cvliTa da sxva. 

amaze masw.-i sagangebod gaamaxvilebs yuradRebas. 

maSasadame, Cven cifrebis weras gvian viwyebT. am dros 

klasSi masw.-ma dafaze unda daxazos Zalian didi ujredebi, 

daaxloebiT 20-santimetrianebi (raTa moswavleebma kargad 

dainaxon yvelaferi). am ujredebSi unda daweros axali cifri, 

yuradReba gaamaxvilos yfmjt! np[sbpcjt! njnbsUvmfcb{f da 

djgsjt! dbmlfvm! obxjmfc{f. Semdeg moswavleebi iwyeben 

TavianT rveulebSi imave cifris weras nimuSis mixedviT. am 

dros masw.-i Camoivlis merxebs Soris, Seamowmebs, Tu ramdenad 

kargad icaven moswavleebi xelis moZraobis mimarTulebas. 

roca klasi iswavlis yvela cifris weras, klasSi weris 

sagangebo samuSaoebi aRar Catardeba. masw.-i mxolod xandaxan 

merxebs Soris CamovliT amowmebs, Tu ramdenad sworad asruleben 

moswavleebi xelis moZraobas saklaso muSaobisas. garda 

amisa, is SeniSvnas miscems moswavles, romelic jRabnis, ar indomebs 

da sufTad ar wers. weraze muSaoba amiT ifargleba. 

danarCeni meore klasSi daixveweba, roca bavSvebis TiTebis 

moZraobis koordinaciis unari metad iqneba momwifebuli. 

TfojTwob/ albaT, ukeTesi iqneba, Tuki I klasSi bavSvebi 

fanqriT daweren xolme. saqme isaa, rom maT xSirad SeeSlebaT 

da fanqriT dawerilis Sesworeba gacilebiT ufro mosaxerxebelia, 

vidre kalmistriT dawerilisa.

- 14 - 

§ 5. Pprograma, misi ZiriTadi Temebi 

ariTmetika: 

1. raodenoba, misi ucvleloba grovis ganlagebis SecvliT. 

2. raodenobaTa Sedareba, maTi met-nakleboba da toloba. 

utolobani. 

3. Tvla (29-is farglebSi). ricxvTa saxelwodebani. 

4. jufTebi da jufTebiT Tvla. 

5. Sekreba (20-is farglebSi). 

6. gamokleba (20-is farglebSi). 

7. cifrebi. ricxvTa, moqmedebaTa, toloba-utolobaTa 

maTematikuri Canawerebi. 

algebra da gamoyenebiTi maTematika: 

8. ariTmetikul moqmedebaTa umartivesi algebruli Tvisebani: 

SesakrebTa gadanacvleba, Sekrebisa da gamoklebis kavSiri. 

9. mravalmoqmedebiani gamoTvlebi (ricxvTa kibeze an zepirad). 

10. sayofacxovrebo amocanebi. 

11. martivi sqemebi (algebruli, logikuri). 

12. cxrili. 

13. rigobiTi ricxviTi saxelebi. 

14. Tanmimdevroba sivrcesa da droSi. ricxvTa Tanmimdevrobani. 

15. ZiriTadi sivrciT-droiTi mimarTebani. 

geometria, xazva da gazomva: 

16. xazebi: swori (monakveTi), texili, mrudi; sxvadasxvagvari 

nakvTebi; kubi. 

17. naswavli nakvTebis xazva. 

18. mravalkuTxedebi, maTi saerTo gverdebi da saerTo wveroebi. 

19. nakvTis sazRvari, Sekruli da gaxsnili xazebi. 

20. sigrZe da gazomva; dm da sm. texilis sigrZe. 

kombinatorika da logika: 

21. kombinatorika: yvela SesaZleblobis amowurva. 

22. dajgufeba (klasifikacia). logikuri kanonzomierebani. 

23. logikuri da maTematikuri terminebi. 

24. logikuri mimarTebani da logikuri msjeloba.

- 15 - 

am 24 Temis sakiTxebi gadanawilebulia tbn![jsjUbe!yb{Tj; 

pirveli, wamyvani xazi _ ariTmetika; 

meore xazi _ algebra, gamoyenebiTi maTematika, kombinatorika; 

mesame xazi _ geometria, xazva, gazomva, dajgufeba. 

logika ki SeWrilia samive xazSi, sakmaod Zlierad. Cveni 

programisa da meTodikis erTerTi maxasiaTebeli swored esaa 

_ upirvelesi mniSvneloba eniWeba moswavlis azrovnebas, logikas. 

mravali sakiTxis uCveulo damuSaveba da TviT programis 

uCveulobac (magaliTad, is, rom cifrebi Semogvaqvs gvian, weris 

swavlebasac viwyebT gvian da sazogadod, bavSvebs cotas 

vawerinebT) gamowveulia swored imiT, rom CvenTvis mTavaria 

namdvili azrovneba da logika. amgvarad, logika _ samive xazis 

gamWoli mTavari ZarRvia, Tumc ufro xSirad _ faruli. 

yb{j _ esaa erTianad agebul, erTmaneTze daSenebul 

programul sakiTxTa Tanmimdevroba, Tanwyoba. igi ramdenime 

Temis sakiTxebs aerTianebs. xazi dayofilia Tavebad (TviTeuli 

Tavi bolovdeba SejamebiTa da gameorebiT), Tavi ki erTianad 

gadanomrili sagakveTilo monakveTebisagan Sedgeba. 

programis danawileba sam xazad sakmaod pirobiTia, radganac 

xazebi xSirad ukavSirdeba erTmaneTs da esa Tu is sakiTxi SeiZleboda 

sxva xazSic yofiliyo CarTuli. pirveli da meore xazi 

TavisTavadaa mWidrod urTierTdakavSirebuli. garda amisa, 

ariTmetikul TemebSi Tavidanve gamoiyeneba geometriuli nakvTebi. 

xolo geometriaSi Tavidanve saWiroa ricxvebi da Tvla. 

cxrilis Tema ganuyofladaa dakavSirebuli geometriasTanac da 

ariTmetikasTanac. gazomvis Temis dasasrulic erTdroulad 

erwymis geometriasac da ariTmetikasac. amgvarad iqmneba erTiani 

maTematikis safuZveli. sami xazi ki saWiro gaxda imisaTvis, raTa 

kargad dagegmiliyo maTematikis gakveTilebi (saqme isaa, rom 

TiTo gakveTilze muSavdeba sami sagakveTilo monakveTi, TiTo 

xazidan TiTo _ amis Sesaxeb dawvrilebiT _ ix. $ 7). 

sazogadod, pirveli xazi wamyvania im azriT, rom maszea 

agebuli mTeli programa da misi damuSaveba da gaazreba qmnis 

mTeli programis Seswavlis safuZvels. magram es araviTar

- 16 - 

SemTxvevaSi ar gulisxmobs imas, rom pirveli xazi ufro 

mniSvnelovania, vidre, magaliTad, mesame. azrovnebis ganviTarebisaTvis 

swored mesame xazia yvelaze mniSvnelovani! 

Cveneuli programis erTerTi siaxle swored geometrialogikis 

xazis Zlieri damuSavebaa, Tanac iseTi, romelic 

bunebrivadaa Serwymuli ariTmetikis xazTan. 

dagegmil gakveTilTa saerTo raodenobaa 135-140. maTematikis 

gakveTili kviraSi xuTjer tardeba, weliwadSi sul daaxloebiT 

155-160 gakveTili unda Catardes (gacdenebisa da 

uqmeebis gaTvaliswinebiT). ese igi, Cven daugegmavad vtovebT 

daaxloebiT 15-20 gakveTils. isini ori ramisTvisaa saWiro. 

jer erTi, Tuki masw.-i SeamCnevs, rom klass (saSualo moswavleebs) 

gauWirdaT raime sakiTxi, is erTi gakveTiliT SeaCerebs 

winsvlas dagegmil saswavlo programaSi da am erT gakveTils 

dauTmobs dagrovebuli Zneli sakiTxebis gameorebasa 

da ganmtkicebas. meorec, sasurvelia, rom saswavlo wlis 

bolosTvis morCes 5-10 gakveTili mainc, raTa moxdes 

umniSvnelovanesi naswavli sakiTxebis saerTo gameoreba, ganmtkiceba 

da ganviTareba, erTmaneTTan dakavSirebulad. am dros 

klasi msr-s mihyveba da, saWiroebisamebr, iyenebs Tvalsa- 

Cinoebis rveulsa Tu saklaso TvalsaCinoebas. 

ise ki masw.-i unda moeridos programaSi CamorCenas, 

zedmetad ar unda SeaCeros winsvla. 

§ 6. ra unda icodes moswavlem saswavlo wlis bolos 

sakiTxebs CamovTvliT meore, maRali donis programis 

mixedviT. meore donis programis sakiTxebis kargi codna 

moeTxoveba mxolod Zlier moswavleebs. amitom iqve 

kvadratul frCxilebSi [ ] mivuTiTebT, Tu romelia SedarebiT 

Zneli sakiTxi da ese igi, SeiZleba gauWirdes saSualo 

moswavles. aseve, imave frCxilebSi vuTiTebT pirveli donis 

programis moTxovnas, Tuki igi gansxvavebulia. Tuki sakiTxs 

es ara aqvs miwerili, ese igi, igi sruli moculobiT Sedis

- 17 - 

pirveli donis programaSi anu savaldebulo minimumSi, 

romelic yvela moswavles moeTxoveba. 

cxadia, sakiTxis dasaxeleba mxolod miaxloebiT warmodgenas 

Tu Segviqmnis imaze, Tu konkretulad rogoria programis 

moTxovna. konkretuli moTxovnebi ki Cans im amocanebidan, 

romlebic mocemulia msr-s ukanasknel, Semajamebel 

nawilSi (daaxloebiT, ukanaskneli 10-15 gakveTili, sadac 

axali sakiTxebi aRar Semodis). 

Cveneul saxelmZRvaneloebSi mravladaa kompleqsuri amocanebi 

_ romelTa amosaxsnelad ramdenime sul sxvadasxva 

moqmedebis Catarebaa saWiro (magaliTad, daTvla, gazomva 

da cxrilis Sevseba; anda, amaTuim mravalkuTxedis daxazva, 

misi umoklesi gverdis moniSvna da masSi raimes Caxazva). 

magram saqme isaa, rom es kompleqsuroba Tanmimdevrulia, 

anu Sedgeba ramdenime nabijisagan, romlebic cal-calke sruldeba. 

moswavlem jer mxolod pirvel nabijs unda miaqcios 

yuradReba, Seasrulos igi, Semdeg daiviwyos da momdevnoze 

gadavides, da ase Semdeg, ese igi, mas ar uwevs erTdroulad 

ramdenime ramis keTeba (rac Zalian Zneli iqneboda). 

arsebiTad, esaa amocanebi algoriTmis (instruqciis) Sesrulebaze. 

amgvari amocanebi Zalian mniSvnelovania, radgan xels uwyobs 

moswavlis azrovnebis mowesrigebas da saSemsruleblo 

unarCvevaTa ganviTarebas. miT umetes, rom amgvari amocanebis 

instruqciaSi (algoriTmis aRweraSi) CarTulia logikuri 

kavSirebi da kvantorebi: Uvlj-! bo-! spnfmjnf-! fsUfsUj-! 

fsUbefsUj-!!zpwfmj... 

mTavaria, moswavlem gaigos, rom araa saWiro yvelaferze 

erTad fiqri: Seasrule jer erTi, Semdeg es daiviwye da 

gadadi meoreze, Semdeg esec daiviwye da gadadi mesameze... ... 

Tanac, rogorc yvela sxva saxis amocanebis jaWvedi, esec 

iwyeba jer Zalian advili, sul ornabijiani algoriTmebiT. 

uaRresad mniSvnelovania, rom I klasis bolosTvis moswavles 

SeeZlos umartivesi logikuri msjelobis aramarto wakiTxva 

da gaazreba, aramed agreTve zepirad Catarebac, logiku-

- 18 - 

ri kavSirebisa da kvantorebis gamoyenebiT (ix. 35-e sakiTxi 

qvemore CamonaTvalSi). 

sakiTxTa nusxaSi yvelgan igulisxmeba ocis farglebi, 

garda im SemTxvevebisa, sadac sxva farglebia miTiTebuli. 

1. TvalsaCinod mocemul nebismier raodenobaTa Sedareba 

dawyvilebiT; raodenobis ucvleloba grovis gafarToeba- 

Seviwroebis Semdeg. 

2. ocdacxris farglebSi: ricxvSi aTeulebis raodenoba, 

naSTi da amis kavSiri ricxvis cifrul CanawerTan [cota 

Znelia]; ricxvTa qarTuli saxelwodeba da Cawera; ricxvTa 

rigi da ricxvTa Sedareba; Tvla da ukuTvla; Svidi-rva 

ricxvis [xuTi-eqvsi ricxvis] dalageba matebis an klebis 

mixedviT; xuTi-eqvsi Tanmimdevruli Sekreba-gamoklebis 

Sesruleba ricxvTa kibeze. 

3. wakiTxva da wera niSnebisa: cifruli Canawerebi 0-dan 

29-mde; + _ = > < · (cali). jamisa Tu sxvaobis 

tolobisa da utolobis wakiTxva da Cawera. 

4. zepiri Sekreba-gamokleba (nuliani Sedegebis magaliTebiTurT) 

[aTis farglebSi _ mtkiced, ocis farglebSi _ garda 

Znelebisa]; ramdeni aklia ricxvs aTamde an ocamde; Sekrebagamoklebis 

magaliTebis cifruli Cawera. 

5. TvalsaCinod mocemuli grovis dalageba jufTebad _ 

mocemuli Canaweris [sityvieri dasaxelebis] mixedviT; dalagebuli 

jufTebis Sesabamasi cifruli gamosaxulebis Cawera; 

am dalagebis gadawyoba sxva jufTebad; raodenobis daTvla 

sxvadasxva jufTebad (naSTian-unaSTo) da Sedegebis Camowera 

erT svetSi [Znelia]. 

6. SesakrebTa gadanacvlebadoba _ terminTa gareSe, mxolod 

konkretuli SemTxvevebisaTvis; ramdeni aklia ricxvs meore 

ricxvamde sxvaobasTan dakavSirebiT. 

7. eqvsi-Svidi Sekreba-gamoklebis Sesruleba ricxvTa 

kibeze; oTxi-xuTi [ori-sami] Sekreba-gamoklebis Sesruleba 

zepirad [Znelia]. 

8. ricxvTa Tu sigrZeTa gadideba da Semcireba.

- 19 - 

9. jamisa da sxvaobis Sedareba ricxvTan an erTmaneTTan, 

maT Soris sayofacxovrebo amocanebSi; jamebisa an sxvaobebis 

erTmaneTTan Sedareba angariSis gareSe, roca erTerTi 

komponenti erTidaigivea _ swrafad [nela]. 

10. advili ormoqmedebiani sayofacxovrebo amocanebi 

Sekreba-gamoklebaze _ ZiriTadi saxeebi: ori paraleluri an 

Tanmimdevruli Sekreba an gamokleba sayofacxovrebo amocanebSi 

[Znelia isini, romlebSic saWiroa ricxvisaTvis jamis mimateba, 

ufro Znelia sxvaobis mimateba]; ucnobi Sesakrebis, saklebis 

an maklebis moZebna, maT Soris sayofacxovrebo amocanebSi. 

11. umartivesi sammoqmedebiani sayofacxovrebo amocanebi: 

sami Tanmimdevruli SekrebiT an advili gamoklebiT [Znelia]. 

12. sayofacxovrebo amocanis pirobis dakvirvebiT gaazreba. 

13. Sekreba-gamoklebis cxrilebis Sevseba; naswavli an 

kargad nacnobi masaliT: sxvadasxvamonacemebiani cxrilebidan 

monacemTa amokiTxva da cxrilebis Sevseba. 

14. TvalsaCino ganlagebisas, gaferadebisas da sxva amgvar 

SemTxvevebSi yvela SesaZleblobis amuwurva, roca maTi 

raodenoba ar aRemateba 6-s [4-s _ Znelia]. 

15. samidan ocamde ricxvebis daSla or-or SesakrebTa 

jamebad _ yvela SesaZleblobis amowurviT; yvela 

SesaZleblobis amowurviT: mocemuli sami ricxvisagan yvela 

SesaZlo erTmoqmedebiani tolobis Sedgena. 

16. umartivesi sqemebis mixedviT moqmedeba, umartivesi 

sqemebis daxazva. es sqemebi moicavs: ricxvebze orsafexurian 

moqmedebebs, ricxvTa met-naklebobas, naswavli tipis 

sayofacxovrebo amocanebs, Sesabamisobas, manZilebsa da 

moZraobas, rigobiT ricxvebsa da Tanmimdevrobas. 

17. sivrciTi mimarTebani: didi \ patara (gansxvavebiT 

mimarTebisagan bevri \ cota), zemoT \ qvemoT, win \ ukan, 

marj-vena \ marcxena [SeiZleba zogs eSlebodes], zemo marjvena 

\ qvemo marcxena da sxva [Znelia], maRali \ dabali, grZeli \ 

mokle, sqeli \ Txeli, msxvili \ wvrili, farTo (ganieri) \ 

viwro [ukanaskneli sami SeiZleba erTmaneTSi ereodeT].

- 20 - 

18. droiTi mimarTebani: ufrosi \ umcrosi, adre \ gvian, 

jer \ Semdeg; sivrciT-droiTi mimarTebani moZraobisas. 

19. Tanmimdevroba sivrcesa da droSi. ricxvTa 

Tanmimdevrobani. 

20. rigobiTi ricxvebi pirvelidan ocdamecxremde 

[meocemde] da maTi dakavSireba naswavl sivrciT-droiT 

mimarTebebTan; martivi Tanmimdevroba sivrcesa da droSi. 

21. nakvTebis dajgufeba da saxelwodebani: wre, rgoli, 

kubi, wertili, monakveTi, kvadrati, marTkuTxedi, samkuTxedi, 

oTxkuTxedi, ... , aTkuTxedi, ... mravalkuTxedi. 

22. kubis wveroebi. mravalkuTxedis gverdebi da wveroebi. 

maT raodenobaTa toloba. 

23. mravalkuTxedTa saerTo gverdebi da saerTo wveroebi. 

24. xazebi: texilebi da mrudebi; gaxsnili da Sekruli xazebi; 

gaxsnili xazebis boloebi (bolo wertilebi), monakveTis 

boloebi. xazebis gadakveTis wertilebi da gadamkveTi xazebi. 

ori wertilis SeerTeba sxvadasxvagvari xazebiT. 

25. naswavli nakvTebis xazva swori da naxvretebiani 

saxazavebiT. 

26. nakvTis Siga da gare wertilebi. nakvTis sazRvari. 

27. sigrZis Tvisebebi _ simetriuloba, tranzituloba da 

adiciuroba (mxolod praqtikulad _ ese igi, am terminTa da 

zogadi Camoyalibebis gareSe); gazomva sigrZis or-ori erTeuliT: 

decimetr-santimetriT, mtkavel-TiTiT, nabij-terfiT. 

28. didi metriani saxazavisa da patara saxazavis danayofebs 

Soris manZilTa advili gamoTvla; gazomva am saxazavebiT. 

29. mocemuli sigrZis mqone monakveTebis xazva (roca 

sigrZe santimetrebis mTel ricxvs Seicavs, milimetrebiani 

naSTis gareSe); anda amgvari gverdebis mqone mravalkuTxedebisa 

da texilebis xazva. 

30. uadvilesi amocanebi manZilsa an moZraobaze, romelTa 

amosaxsnelad saWiroa erTi erTeuliT gazomil manZilTa 

Sekreba-gamokleba. 

31. kombinatorika geometriaSi _ ramdeni samkuTxedis,

- 21 - 

kvadratis an sxva nakvTis danaxvaa SesaZlebeli naxazze, yvela 

SesaZleblobis amowurviT [Znelia] _ roca maTi raodenoba ar 

aRemateba 6-s [4-s]; an radara nakvTebis danaxvaa SesaZlebeli 

naxazze, yvela SesaZleblobis amowurviT _ roca maTi 

raodenoba ar aRemateba 6-s [4-s _ Znelia]. 

32. naswavli mravalkuTxedebis dayofa sxvadasxva mravalkuTxedebad; 

ori mravalkuTxedisagan axalis Sedgena [Znelia]. 

nacnobi sagnebis dajgufeba ramdenime jgufad ori [erTi] Tvalsa- 

Cino niSanTvisebis mixedviT [mxolod TvalsaCino masaliT]. 

33. logikuri kanonzomierebani: geometriul, ariTmetikul, 

grafikul Tu sayofacxovrebo masalaze [Znelia, roca masala 

sityvieradaa mocemuli]. mravalgvari amocanebi kanonzomierebis 

darRvevasa an mis gagrZelebaze [zogi Znelia]. 

34. amocanebi, romlebSic erTdroulad monawileobs am 

programidan sami [ori] sakiTxi. amocanebi sxvadasxva 

saprogramo sakiTxebis erTmaneTTan dakavSirebaze. 

35. terminebi: ìmdenive~, `Sekreba~, `mimateba~, `Sesakrebi~, 

`jami~, `gamokleba~, `saklebi~, `maklebi~, `sxvaoba~, `da 

kidev~, `naSTi~, `toloba~, ùtoloba~, `metia~, `naklebia~, 

`Sedareba~, `daxazva~, `gazomva~, `sigrZe~; `garda~, `mxolod~, 

èrTnairi~, `toli~, `yvela~, `yvelaze~ (meti, didi, mokle da 

sxva), `mravali~, `TviTeuli~, `yoveli~, `zogierTi~, `TiTo- 

TiTo~, `ramdenime~, èrTerTi~, àrcerTi~, ùfro ... vidre~, 

`da~, àn~, `romelime~, ìgive~, `Tuki~, `maSin~, `ratom~, 

ìmitom rom~, àmitom~, `radgan~, àmocanis piroba~, àmoxsna~. 

mxolod konkretuli saxis amocanebis konteqstSi: 

`sapirispiro~, `Sesabamisi~, `SesaZlebloba~, `dajgufeba~, 

`kanonzomiereba~. cxadia, yvelaferi mxolod TvalsaCino 

masaliT keTdeba. am terminebs ver gaveqceviT, masw.-i maT 

Tavidan ixmars yovelgvari axsna-ganmartebis gareSe, saWiroebis 

mixedviT, magram yovelTvis _ konkretul da TvalsaCino 

viTarebaSi. moswavleebi am sityvebis mniSvnelobas mainc ver 

Caswvdebian, I klasSi es arcaa saWiro. ubralod, maT unda 

icodnen, ra moeTxovebaT, roca amocanis piroba moiTxovs: àr

- 22 - 

gamogrCeT arcerTi SesaZlebloba!~ an àrRvevs kanonzomierebas~ 

da sxva. amas ki sakmaod male aiTviseben. 

§ 7. ZiriTadi saprogramo xazebis aRwera _ 

Sesabamisoba saxelmwifo saswavlo gegmasTan 

jer Cven mokled, sqematurad aRvwerT samive xazs mTlianobaSi 

_ raTa masw.-s mTliani warmodgena Seeqmnas imaze, Tu ra 

TanmimdevrobiT da rogor viTardeba am xazis Sinagani azri. 

Semdeg ki mocemulia TviTeuli gakveTilis gegma-konspeqti, 

sakmaod dawvrilebiT _ is tipuri SekiTxvebi, romlebic unda 

dasvas masw.-ma, raTa klasi miiyvanos sakiTxis arsamde. iqve aRwerilia 

rogorc saTanado TamaSebi, ise msr-dan saSinao davalebisa 

Tu saklaso samuSaos davalebaTa mokle ganxilva. 

xazi dayofilia Tavebad, Tavi _ sagakveTilo monakveTebad. 

Tavi moicavs erT an or axal sakiTxs. magram, rogorc iTqva 

zemore, am axali sakiTxis swavleba erTbaSad, mzamzareulad 

ar xdeba. igi danawilebulia imdenad mcire safexurebad, 

romlebsac saSualo moswavlec ki TiTqmis damoukideblad 

gaivlis (Tuki is, cxadia, masw.-ma klasSi saTanadod amuSava) _ 

radganac TviTeuli es safexuri aqtiuradaa Semzadebuli wina 

safexurebiT. es mcire safexuri muSavdeba erTi gakveTilis 

daaxloebiT mesamed-meoTxed droSi. esaa erTi sagakveTilo 

monakveTi. gakveTilis sxva sagakveTilo monakveTebi eTmoba 

danarCeni ori xazis amdagvarsave mcire-mcire safexurebs, 

agreTve TamaSsa da saSinao davalebis erTobliv garCevas. 

amgvarad, TviTon xazia ise agebuli, rom yovel 

sagakveTilo monakveTze mxolod erTi mcireodeni siaxle 

Semodis da isic _ jerovnad Semzadebuli. magram gakveTilze, 

garda am axali sakiTxis ganxilvisa, aucileblad unda moxdes 

wina, naswavli sakiTxebis gameorebac. zogjer axali sakiTxi 

imdenad advilia, rom Zalian mcire dro sWirdeba; xSirad ki 

axali sakiTxi mxolod sul umniSvnelod gansxvavdeba wina 

naswavli sakiTxisagan. es ganzraxaa ase dagegmili, raTa xSirxSirad 

moxdes naswavlis gameoreba, Tanac ara mxolod

- 23 - 

ubralo gameoreba, aramed gameoreba sxvadasxva kuTxiT. Tu 

rogor _ es mocemulia gegma-konspeqtebSi. 

gegma-konspeqtebi gakveTilebis mixedviTaa Sedgenili. 

yoveli gakveTilis gegma-konspeqti Seicavs sami sxvadasxva 

xazis nawyvets, agreTve TamaSis aRwerasa da saSinao davalebis 

ganxilvas. amitom gegma-konspeqti ver gagavagebinebs imas, Tu 

rogor muSavdeba esa Tu is Tema mTlianobaSi. 

masw.-ma jer mTlianobaSi unda gaiazros, Tu rogor aigeba 

TviTeuli xazi, rogor muSavdeba misi ZiriTadi sakiTxebi, 

rogoria ganviTarebis Sinagani logika. 

Tanac, sakiTxebis yovel jgufs miwerili aqvs misaRwevi 

miznebi saxelmwifo programidan, romlebsac emsaxureba igi. 

magaliTad, 1.1 emsaxureba I.4-is miRwevas. 

qjswfmj!yb{j!`!bsjUnfujlb! 

1.1. raodenobaTa met-naklebobis garkveva dawyvilebiT: 

mxolod TvalsaCinod mocemuli nivTebis grovebze, utolobis 

niSanTa gareSe. bevri \ cota. ìmdenive~. raodenobaTa toloba. 

dawyvileba, roca erTi grova uZravia; Semdeg _ roca 

orive grova uZravia _ xazebiT SeerTebis gziT. I.4 

2.1. raodenobani erTidan eqvsamde: mxolod TvalsaCinod 

mocemuli nivTebis grovebiT, cifrul niSanTa da CanawerTa 

gareSe. ZiriTadi sayrdenia dominosebri TvalsaCinoeba. mTavaria 

imis gaazreba, rom raodenoba ar icvleba grovis ganlagebis 

SecvliTa da misi gafarToebiT. I.1, I.4 

3.1. moqmedeba Sekreba. mimateba. zepiri Sekreba 6-is farglebSi. 

SesakrebTa gadanacvlebadoba (am terminTa gareSe). I.3 

4.1. moqmedeba gamokleba. zepiri gamokleba 6-is farglebSi, 

nivTTa grovebze, gamoklebis niSnisa da cifrebis gareSe. 

ZiriTadi sayrdenia dominosebri TvalsaCinoeba. 6-is farglebSi 

jamTa da sxvaobaTa angariSi da pirveli damaxsovreba. I.3 

5.1. raodenobani 7-dan 9-mde: mxolod TvalsaCinod mocemuli 

nivTebis grovebiT, cifrul niSanTa da CanawerTa gareSe. 

am raodenobaTa grovebi da Sedareba. zepiri Sekreba-gamokleba

- 24 - 

9-is farglebSi, nivTebiT, moqmedebis niSnebisa da cifrebis 

gareSe. ricxvTa rigi: erTi, ori, ... , rva, cxra. ukuTvla. I.1,2,3. 

6.1. 1-dan 9-mde cifrebis wakiTxva. sxva ariTmetikuli niSnebi: 

metobisa, naklebobisa, tolobisa, Sekrebisa, gamoklebisa. 

wers mxolod masw.-i (carciT dafaze), moswavleebi klasSi 

moZravi (magnituri) cifrebiT `weren~ dafaze. Tan xdeba 

naswavlis gameoreba, ukve cifrul CanawerTa doneze. I.1, I.3 

7.1. terminebi `Sesakrebi~ da `jami~ _ mxolod TvalsaCino 

CvenebiT. ricxvi nuli, cifri nuli. ricxvisTvis nulis 

mimateba da ricxvis nulisTvis mimateba. I.3 

8.1. terminebi `saklebi~, `maklebi~ da `sxvaoba~ _ mxolod 

TvalsaCino CvenebiT. ricxvi aTi _ dominosebri TvalsaCinoebiT, 

cifruli Canaweris gareSe. zepiri Sekreba-gamokleba aTis 

farglebSi. ricxvisTvis nulis gamokleba. ricxvisTvis Tavis- 

Tavis gamokleba. Canaweri `10~ jer ar Semogvaqvs, radganac 

jer ver vaCvenebT imas, Tu ra saWiro gaxda ricxvis Cawera 

ori cifriT. aqamde ricxvebs xom TiTo cifriT vwerdiT! xom 

SeiZleboda, àTic~ Cagvewera raime erTi niSniT!? Canawerebs 

`10~, `11~, `12~ da ase Semdeg SemovitanT mogvianebiT, roca 

mzad viqnebiT maT dasasabuTeblad. I.1, I.3, I.4 

9.1. TvalsaCinod mocemul raodenobaTa daTvla jufTebiT 

_ unaSTo SemTxvevebi (mxolod zepirad, saTanado CanawerTa 

gareSe). terminebi `yoveli~, `TviTeuli~. zogad termins 

`jufTi~ klasSi arcerTxel ar vaxsenebT, igi mxolod masw.-is- 

Tvisaa. klasSi vityviT mxolod konkretulad: oTxeuli, aTeuli, 

cxraeuli, wyvili da sxva _ romelic gvWirdeba. xdeba 

TvalsaCinod mocemuli raodenobis daTvla jufTebad: `daTvale 

oTxeulebad! daTvale wyvilebad! daTvale aTeulebad!~ sa- 

Wiro pasuxis nimuSi: èqvsi cali oTxeuli~. I.1, I.4 

10.1. unaSTo jufTebis Cawera da wakiTxva. jufTebis dawyoba 

winaswar mocemuli Canaweris mixedviT (ese igi, moqmedebaTa 

Cawera maTematikuri niSnebiT). erTeuli, anda erTi cali juf- 

Ti, viTarc jufTebiT daTvlis kerZo SemTxvevebi. 

I da II klasebSi niSani · aRniSnavs mxoloddamxolod sit-

- 25 - 

yvas `cali~. araviTar SemTxvevaSi ar SeiZleba gamravlebis 

xseneba, namravlis gamoTvla da sxva! arc imis gaazrebaa 

saWiro, rom 3 · 4 raodenobiT tolia 4 · 3-isa. I.1, I.4 

11.1. naSTiani jufTebi. naSTi. naSTiani SemTxvevebis dawyoba, 

Cawera da wakiTxva. kerZo SemTxveva _ naSTi 0. gaiazreba Zneli 

sakiTxi: naSTi ar SeiZleba iyos jufTze meti. erTidaimave 

grovis daTvla sxvadasxva jufTebiT, naSTian-unaSTod. I.4 

12.1. Tvla aTeulebiT, viTarc jufTebiT Tvlis kerZo SemTxveva. 

TvalsaCinod mocemul raodenobaSi aTeulTa gamoyofa. 

ricxvebi 10-dan 20-mde, maT qarTul saxelwodebaTa Tanmimdevruli 

gaazreba (cifrul CanawerTa gareSe!): `T-or-meti~, `sammeti~ 

anu `cameti~, `T-oTx-meti~, `T-xuT-meti~ da ase Semdeg. 

zepiri Sekreba-gamokleba ocis farglebSi _ dominoseburi 

TvalsaCinoebiT. arsebiTia, rom am droisaTvis moswavleebma 

ukve kargad ician `-iT metoba~ da `-iT nakleboba~ _ `sayofacxovrebo 

amocanebis~ xaziT. Sekreba-gamokleba àTeulze 

gadausvlelad~, rac ar Cans dominosebur TvalsaCinoebaze 

(11+2, 18_3 da misT.), _ jer ar ganixileba. I.1, I.3, I.4 

13.1. 10-dan 20-mde ricxvebis cifruli Cawera. 21-dan 29mde 

ricxvebi da maTi cifrulad Cawera. cifrul CanawerTa 

gaazreba da dasabuTeba. Sekreba-gamoklebis naswavli magali- 

Tebis cifruli Cawera da gameoreba. 21-dan 29-mde ricxvebisa 

da maTi cifruli Caweris swavla xels uwyobs ricxvis aTobiTi 

cifruli Canaweris gaazrebas (mxolod 20-is farglebSi 

es uaresad Cans). amasTan, 30-mde ricxvebi ar Seicavs araviTar 

damatebiT siZneles, 20-mde ricxvebTan SedarebiT. I.1, I.2, I.3 

14.1. Sekreba-gamokleba 20-is farglebSi àTeulze gadausvlelad~ 

_ dominosebri TvalsaCinoebiT (advilia). 20-is daSla 

SesakrebTa jamebad _ òceulis TvalsaCinoebaze~ damyarebiT. 

ramdeni aklia ricxvs aTamde. ramdeni aklia ricxvs ocamde. 

15.1. ZiriTadi naswavli sakiTxebis gameoreba, ganmtkiceba da 

ganviTareba _ nusxa ix. $ 6-Si. I.1, I.2, I.3 

nfpsf!yb{j!`! 

bmhfcsb-!hbnpzfofcjUj!nbUfnbujlb-!lpncjobupsjlb!

- 26 - 

1.2. umartivesi sivrciTi mimarTebani: didi \ patara (gansxvavebiT 

bevri \ cotasagan), zemoT \ qvemoT, win \ ukan, 

Soris, maRali \ dabali. ganlageba simaRlis mixedviT. 

yvelaferi _ TvalsaCinod. mimarTeba SigniT \ gareT. I.7 

2.2. Semdgomi sivrciTi mimarTebani: sqeli \ Txeli, farTo 

\ viwro, msxvili \ wvrili. rigobiTi ricxvebi erTidan eqvsamde 

(mxolod sityvierad). maTi dakavSireba umartives sivrciT 

mimarTebebTan. advili sivrciTi mimarTebani warmodgeniT. 

gansxvavebiT Tavi 1-is mimarTebaTagan, moswavleebs erTmaneTSi 

aerevaT xolme am Tavis zogi mimarTeba. zogs SeiZleba gauWirdes 

agreTve sivrciT mimarTebaTa garkveva warmodgeniT. I.7 

3.2. raodenobis mateba an kleba umartives sayofacxovrebo 

amocanebSi. umartivesi sayofacxovrebo amocanebi raodenobis 

momatebasa da Sekrebaze; raodenobis moklebasa da gamoklebaze. 

TvalsaCino raodenobaTa `-iT metoba~ da `-iT nakleboba~. I.2 

4.2. sivrciTi mimarTeba marjvniv \ marcxniv. Soris. misi 

dakavSireba naswavl rigobiT ricxvebTan. ormagi sivrciTi 

mimarTebani: wina zemoTa, ukana qvemoTa da misTanani. I.7 

5.2. rigobiTi ricxvebi: meSvide, merve, mecxre. ricxvTa 

kibe da Sekreba-gamokleba masze. ricxvis -iT Semcireba an 

gadideba da ricxvTa kibeze Sesabamisi wanacvleba. I.2 

6.2. Sekrebis, gamoklebis cxrilebi da maTi Sevseba. I.3 

7.2. rigobiTi ricxvi meaTe da misi dakavSireba sivrciT 

mimarTebebTan. kombinatorikis uadvilesi sawyisebi: yvela SesaZleblobis 

amowurva, roca maTi raodenoba ar aRemateba 6-s 

(mxolod TvalsaCino masalaze). `TiTo-TiTo~. I.2 

erTidan eqvsamde ricxvebis Sesabamis dominosebur wyobaTa 

ganlageba yvela SesaZleblobis amowurviT, roca ucnob 

SesaZleblobaTa raodenoba ar aRemateba 6-s. I.4 

8.2. samidan aTamde ricxvebis daSla SesakrebTa jamebad _ 

yvela SesaZleblobis amowurviT. esaa kombinatorikisa da 

ariTmetikis saerTo sakiTxi, Zalian mniSvnelovani. erToblivi 

Ziebis Semdeg dafaze Camoiwereba yvela SesaZlebloba 

tolobaTa saxiT. klasi Tan kiTxulobs. I.2, I.4

- 27 - 

ormoqmedebiani Sekreba-gamokleba ricxvTa kibeze. `-iT 

metoba~ da `-iT nakleboba~ umartives anu erTmoqmedebian 

sayofacxovrebo amocanebSi. viwyebT ormoqmedebian gamoTvlebs. 

swored mravalmoqmedebian gamoTvlebs aadvilebs ricxvTa kibe, 

Torem erTmoqmedebianebis masze Sesrulebas didi azri arca 

aqvs. klass TandaTan zepiradac vasrulebinebT advil 

ormoqmedebian gamoTvlebs. Secdomisa an gaWirvebis SemTxvevaSi 

kvlav ricxvTa kibes viSveliebT. I.2, I.4 

9.2. jamisa da sxvaobis Sedareba ricxvTan, maT Soris sayofacxovrebo 

amocanebSi. ramdeni aklia ricxvs meore ricxvamde 

sxvaobasTan dakavSirebiT. viTardeba wina Tavis Tema, oRond 

Tan kavSirdeba mocemuli ori ricxvis sxvaobasTan. I.3 

10.2. umartivesi droiTi mimarTebani TvalsaCino 

mocemulobiT: ufrosi \ umcrosi. adre \ gvian. jer \ Semdeg. 

win \ Semdeg. maTi dakavSireba rigobiT ricxvebTan. I.7 

jamebisa da sxvaobebis Sedareba, roca erTerTi komponentia 

(am terminis gareSe!) erTidaigive. es sakiTxi ramdenime wina 

safexurs saWiroebs: jer xdeba mxolod jamebis Sedareba, 

roca pirveli Sesakrebia erTidaigive, da ase Semdeg. Sedarebis 

es sakiTxebi am TavSi mTavrdeba. Tumca, jamebisa da sxvaobebis 

Sedareba saswavlo wlis bolomde grZeldeba, oRond 

ZiriTadad mxolod saSinao davalebebSi. TandaTan moswavleebi 

gaiazreben, rom jamebisa da sxvaobebis Sedarebisas, roca 

erTerTi komponentia erTidaigive, araa saWiro angariSi _ 

isedac Cans, romelia meti. magaliTad, 13 _ 4 > 13 _ 7. 

Tumca, amis Tavidanve swavleba araa saWiro. umjobesia, rom 

moswavlem TviTon iswavlos es mravali savarjiSos 

Sesrulebis Sedegad. I.3, I.4 

11.2. ori paraleluri (ara Tanmimdevruli!) Sekreba an 

gamokleba ormoqmedebian sayofacxovrebo amocanebSi. Tanmimdevrul 

Sekreba-gamoklebas sWirdeba pirveli Sedegis damaxsovreba 

da amitom ufro Znelia, viwyebT jer paraleluri SekrebagamoklebiT. 

am SemTxvevaSi ori moqmedeba erTmaneTisagan damoukideblad 

sruldeba da amdenad, ufro advilia. I.2, I.3

- 28 - 

naswavli droiTi mimarTebani gonebaSi warmodgeniT an 

sqemiT da maTi dakavSireba rigobiT ricxvebTan. I.7 

sxvadasxvamonacemebiani cxrilis amokiTxva da misi Sevseba. 

aqamde gvxvdeboda mxolod samgvari cxrilebi: nakvTebis formaferisa, 

mravalkuTxedebis gverdebisa da wveroebis raodenobisa 

da Sekreba-gamoklebisa. axla Semogvaqvs zogadad, sxvadasxvamonacemebiani 

cxrili (vgulisxmobT, rom am droisaTvis moswavleebma 

ukve sakmaod ician kiTxva). Sekrebisa da gamoklebis 

sxvadasxva cxrilebis Sevseba ki saswavlo wlis bolomde 

grZeldeba, oRond ZiriTadad mxolod saSinao davalebebSi. I.3 

12.2. ori paraleluri Sekreba an gamokleba sayofacxovrebo 

amocanebSi, roca monawileobs sityva ìmdenive~ da, SesaZloa, 

nuli. ori Tanmimdevruli Sekreba an ori gamokleba ormoqmedebian 

sayofacxovrebo amocanebSi. ormoqmedebiani sayofacxovrebo 

amocanebi, romelTa amoxsna moiTxovs ricxvisa da jamis 

Sekrebas an ricxvisa da sxvaobis Sekrebas. I.2, I.4 

13.2. rigobiTi ricxvebi meTerTmetedan meocemde. maTi 

dakavSireba naswavl sivrciT da droiT mimarTebebTan. I.2 

sayofacxovrebo amocanebi, romelTa amosaxsnelad saWiroa 

Sekrebisa an gamoklebis erTi ucnobi komponentis moZebna (am 

terminTa gareSe). zogadi gegma amgvaria: 

1) nivTebze moqmedebebi da Tan mTw-s TvalsaCinoeba; 

2) TvalsaCino sqemebi da Tan mTw-s TvalsaCinoeba; 

3) sayofacxovrebo amocanebi _ romelTa konkretuli Sinaarsi 

aadvilebs sakiTxSi garkvevas, TviTon hkarnaxobs moswavles 

amoxsnis gzas; 

4) `Cavifiqre ricxvi”-s tipis amocanebi; 

5) ucnobi komponentis moZebna ganyenebul viTarebaSi, Sesakreb-sakleb-maklebis 

SiSveli terminebiT. oRond, am terminebSi 

mocemulia mxolod amocana, xolo misi amoxsna kvlav azrianad 

unda moxdes! I.2 

14.2. 10-dan 20-mde ricxvebis daSla SesakrebTa jamebad _ 

yvela SesaZleblobis amowurviT. erToblivi Ziebis Semdeg 

dafaze Camoiwereba yvela SesaZlebloba tolobaTa saxiT.

- 29 - 

klasi Tan kiTxulobs. SesaZleblobaTa Tema sxvagvaradac 

daukavSirdeba ariTmetikul moqmedebebsa da ricxvTa kibes: 

radara sxvadasxva gziT SegviZlia mimateba-gamokleba. maTgan 

avarCevT yvelaze mosaxerxebels, magaliTad: 7 + 9 = 7 + 3 

+ 6 = 10 + 6 = 16. ese igi, miviRebT nawil-nawil Sekrebagamoklebis 

cnobil xerxs. magram, rogorc iTqva I xazSi, es 

Zneli xerxia da moswavleTa nawili mas ver aiTvisebs. 

ormoqmedebiani sayofacxovrebo amocanebi, romelTa amosaxsnelad 

saWiroa ricxvisa da sxvaobis Sekreba; jamisa an sxvaobisaTvis 

ricxvis gamokleba. amiT amowurulia ormoqmedebiani 

sayofacxovrebo amocanebis ZiriTadi saxeebi. I.2, I.3 

15.2. gakveTilebis cxrili. ZiriTadi naswavli sakiTxebis 

gameoreba, ganmtkiceba da ganviTareba _ nusxa ix. $ 5_Si. 

nftbnf!yb{j!`!hfpnfusjb-!yb{wb-!hb{pnwb-!ebkhvgfcb! 

1.3. umTavresi geometriuli nakvTebi da maTi daxazva (maTi 

ganxilvis Tanmimdevroba ganpirobebulia ariTmetikis xaziT: 

raTa ricxvs `sami~ daemTxves samkuTxedi, ricxvs òri~ _ 

orboloiani monakveTi da ase Semdeg). wre. rgoli. kubi da 

misi sxvadasxva xedebi. kvadrati _ kubis cali waxnagi (am 

terminis gareSe). wertili _ kvadratis cali wvero (jer am 

terminis gareSe). monakveTi _ kvadratis cali gverdi (jer am 

terminis gareSe). monakveTis ori bolo _ wertilebi. ori 

wertilis SeerTeba monakveTiT. samkuTxedi. xuTkuTxedi. 

eqvskuTxedi. naswavl mravalkuTxedTa (mravalkuTxedis zogadi 

cnebis gareSe!) gverdebi. maTi daTvla. I.6 

mTavaria, rom moswavleebma amTaviTve gaacnobieron: nakvTis 

raoba ar icvleba misi feris SecvliT an mdebareobis SecvliT 

_ iseve rogorc ariTmetikaSi raodenoba ar icvleba grovis 

ganlagebis SecvliT! (es Sedareba mxolod masw.-isTvisaa da 

ara moswavleTaTvis). 

2.3. sxvadasxvagvari samkuTxedebi, oTxkuTxedebi, xuTkuTxedebi, 

eqvskuTxedebi. maTi daxazva. 

umartivesi dajgufeba, anu erTi TvalsaCino niSanTvisebis

- 30 - 

mixedviT dajgufeba (klasifikacia), Semdeg _ TiTo wevris 

CamatebiT. terminebi `mxolod~, èrTnairi~, `toli~, 

`ramdenime~. naswavli nakvTebis dajgufeba. I.6 

3.3. uadvilesi cxrilebi. cxrilis striqoni, sveti da ujredi. 

cxrilis Sevseba yvelaze advili da TvalsaCino masaliT. 

arsebiTad, amisaTvis saWiroa saganTa daxarisxeba ori Zalian 

TvalsaCino niSanTvisebis mixedviT: ferisa da formis mixedviT. 

oRond, am daxarisxebas sakmaod aadvilebs cxrili. I.6 

SedarebiT rTuli mravalkuTxedebi. iseve rogorc 1-l 

TavSi, maTi Tanmimdevroba ariTmetikis xaziTaa ganpirobebuli. 

varskvlaviseburi (Cazneqili) 7, 8, 9-kuTxedebi. maTi daxazva 

(miaxloebiT). maTi Sedgena sxva mravalkuTxedebisagan. 

4.3. mravalkuTxedis (jer am terminis gareSe) gverdebi da 

kuTxeebi, maTi daTvla. marTkuTxedi, misi daxazva. iseve rogorc 

yvela sxva nakvTi, marTkuTxedic Semogvaqvs TvalsaCino 

CvenebiT, yovelgvar ganmartebaTa gareSe _ magram daxrilad 

daxazulic! sazogadod, gansazRvrebani, magaliTad, kuTxisa an 

marTkuTxedisa, dawyebiT klasebSi uadgiloa. moswavleebs unda 

vaswavloT nakvTebis cnoba (Tanac, yovelgvar mdebareobaSi!) da 

maTi daxazva Tu gazomva. I.6 

viwyebT kanonzomierebis Temas. iqneba jer Zalian advili 

amocanebi logikuri kanonzomierebis darRvevaze _ romeli 

arRvevs kanonzomierebas; Semdeg nakluli kanonzomierebis 

Sevsebaze. termini `kanonzomiereba~ amgvar amocanebSi 

gulisxmobs imas, rom yvela wevri raRaciT erTmaneTs hgavs, 

raRac saerTo aqvT, _ garda erTi wevrisa. swored es erTia 

gamonaklisi, anu `zedmeti~, anu kanonzomierebis damrRvevi. 

meoregvar amocanebSi ki mocemulia wevrTa rigi, romlebsac 

raRac kanonzomiereba aerTianebs. gamonaklisi araa, magram 

saWiroa am kanonzomierebis mixedviT rigis gagrZeleba, carieli 

adgilisa Tu ? niSnis Sevseba. termini `romelime~. I.5 

5.3. kubis wveroebi, maTi daTvla. kvadratis `kuTxis~ 

gaigiveba kubis wverosTan. kvadratisa da, sazogadod, 

mravalkuTxedis wvero _ esaa marTebuli termini, `kuTxis~

- 31 - 

nacvlad. mravalkuTxedis gverdebisa da kuTxeebis raodenobaTa 

cxrili _ cxrilis axali saxe. mravalkuTxedis gverdebisa da 

kuTxeebis raodenobaTa toloba. I.6 

masw.-isTvis aq mravali terminia naxsenebi: waxnagi, 

piramida, prizma, igiveoba da sxva. arcerTi maTgani araa 

moswavleTaTvis. vaswavliT mxolod erT axal termins 

`wvero~. gamovkveTT ukve cnobil terminsac ìgive~. 

kanonzomierebis Tema sagakveTilo monakveTSi aq ukanasknelad 

iqneba. awi igi mxolod saSinao davalebis amocanebSi 

Zalian xSirad, da ase ganviTardeba. I.5 

6.3. kombinatorika geometriaSi _ ramdeni samkuTxedis (an 

sxva nakvTis) danaxvaa SesaZlebeli naxazze, yvela 

SesaZleblobis amowurviT (kombinatoruli amocanebi yvela 

SesaZleblobis amowurviT da termini `SesaZlebloba~ klass 

ukve Sexvda II xaziT). I.7 

nakvTis sazRvari. aTkuTxedi (varskvlaviseburi), misi 

daxatva xeliT, misi gamorCeva sxvagvari nakvTebisagan. iseve 

rogorc sxva mravalkuTxedebi, aTkuTxedic Semogvaqvs zustad 

im gakveTilze, romelzec ariTmetikis xaziT pirvelad 

Semogvaqvs aTeuli da ricxvi aTi. mravalkuTxedi _ zogadi 

terminis Semotana. mravalkuTxedis gverdebi, wveroebi, maTi 

raodenobis toloba. mravalkuTxedis sazRvari. mravalkuTxed- 

Ta dajgufeba: samkuTxedebi, oTxkuTxedebi da ase Semdeg. I.6 

moswavles termini `nakvTi~ ukve mravaljer Sexvda saSinao 

Tu saklaso samuSaoebis Sesrulebisas. masw.-i mas pirdapir, 

TiTiT Cvenebis TanxlebiT ambobda. axlac asea. moswavleebi 

TandaTan SeeCvevian am termins. mis ganmartebas araviTari azri 

ara aqvs. misi zogadi mniSvnelobis gaazreba pirvelklasels 

arc moeTxoveba (gansxvavebiT, magaliTad, cnebisagan `mravalkuTxedi~ 

an `xazi~ _ SemdgomSi iqneba). am TavSi arsebiTad, 

axalia ori cneba: sazRvari da mravalkuTxedi. sxva 

yvelaferi naswavli sakiTxebis mxolod ganviTarebaa, arsebiTi 

siaxlis gareSe. mravalkuTxedi ganazogadebs anu gaaerTianebs 

ukve kargad cnobil samkuTxedebs, oTxkuTxedebs, xuTkuT-

- 32 - 

xedebsa da ase Semdeg. mravalkuTxedi _ mainc rTuli cnebaa 

imis gamo, rom Zalian zogadia. mere TvalsaCino CvenebiT 

varkvevT, wertili mravalkuTxedis SigniT mdebareobs Tu 

gareT. es sakmaod advilad irkveva. wvero? ai wvero ki arc 

SigniTaa da arc gareT. asevea sxva mravali wertilic. yvela 

isini mdebareobs mravalkuTxedis sazRvarze. 

7.3. texili xazi anu texili; misi boloebi da wveroebi; 

texilis Semadgeneli monakveTebis daTvla. Sekruli da 

gaxsnili texilebi da maTi daxazva. wina TavSi vaswavleT 

nakvTis sazRvari. texili Semogvaqvs, viTarc mravalkuTxedis 

sazRvari. Semdeg gamoirkveva, rom igi monakveTebisagan Sedgeba. 

Sekruli xazi iseTia, romlis Signidanac wertils gareT 

ver `gamovasrialebT~ ise, rom sazRvari ar gadavkveToT... I.6 

garda amisa, grZeldeba geometriuli kombinatorikis Tema: 

esa Tu is nakvTi ramdeni SeiZleba davinaxoT naxazze _ yvela 

SesaZleblobis amowurviT (maTi raodenoba ar aRemateba eqvss). 

8.3. geometriuli kombinatorika _ gagrZeleba. mrude xazi 

anu mrudi. wina TavSi vaswavleT texili, viTarc mravalkuTxedis 

sazRvari. axla aseve vaswavliT mrude xazs, rogorc 

saTanado nakvTis sazRvars. I.7 

sigrZis met-nakleboba, `toli sigrZe~ _ datolebiT. misi 

Cawera utolobiT. viwyebT gazomvis Temas. magram sanam TviT 

gazomvas daviwyebdeT, moswavleebma unda gaiazron, Tu ra aris 

sigrZe da ra Tvisebebi aqvs mas. uamisod gazomva meqanikuri 

procesi iqneba. II.10 

9.3. sigrZeTa met-naklebobis Tviseba tranzituloba. gazomva 

Tokis saSualebiT, maT Soris mrude xazisa. zogadi cneba 

`xazi~ (aerTianebs monakveTebs, texilebsa da mrude xazebs). 

Sekruli da gaxsnili xazebi. gaxsnili xazebis bolo 

wertilebi (boloebi). ori xazis gadakveTis wertilebi. ori 

wertilis SeerTeba sxvadasxvagvari xazebiT. am xazebis gazomva 

da maT sigrZeTa Sedareba. monakveTi _ swori da umoklesi 

xazi. garda amisa, viTardeba dajgufebis Temac. II.10 

10.3. gazomva: CxirebiT; CxirebiTa da TiTebiT; mtkaveliTa

- 33 - 

da TiTebiT; nabijiTa da terfiT. yvela am sazomis cvalebadoba 

da arasizuste. zomva didi saxazaviT _ decimetriTa 

da santimetriT. didi saxazavis danayofebis wakiTxva. 

gazomvis TemaSic mivyvebiT bunebriv istoriul gzas. 

moswavle TviTon unda darwmundes, rom saWiroa myari da 

ucvleli sazomi erTeulebi, rom saxazaviT zomva mosaxerxebelia. 

Tumca, yofa-cxovrebaSic Cven xSirad gvWirdeba xolme 

Zveli sazomebis gamoyeneba (mtkavelis, Tokisa da sxva). amitom 

amis codna TavisTavadac saWiroa. 

sigrZis or-ori erTeuli saWiroa imisaTvis, raTa gavzomoT 

pirveli, didi erTeuliT gazomvis Semdeg morCenili naSTi. 

amis saWiroebasac moswavle Tavad Seejaxeba, roca ver ityvis: 

sigrZe 5 Cxiria Tu 6. gazomvis Sedegebs dafaze vwerT. 

Tanafardobas 1 dm = 10 sm jer ar vaswavliT, jerjerobiT 

dm cvlis mtkavels, xolo sm _ TiTs. II.10 

TvalsaCinod mocemuli xazebis dajgufeba: monakveTebi, texilebi 

da sxvebi _ mrudebi. monakveTebi, Sekruli texilebi, 

gaxsnili texilebi, Sekruli mrudebi, gaxsnili mrudebi. I.6 

11.3. dajgufeba gonebaSi warmodgeniT. ori TvalsaCino 

niSanTvisebis mixedviT ramdenime jgufad dajgufdeba kargad 

cnobili sagnebi, oRond maTi mxolod sityvieri dasaxelebisa 

da dafaze Camoweris safuZvelze. dajgufebis Sedegebic 

wamoiwereba dafaze, kvlav sityvierad (am droisaTvis klasma 

ukve sakmaod kargad unda icodes kiTxva). bolos igive keTdeba 

maTematikuri sagnebisaTvis. I.6 

garda amisa, viTardeba gazomvis Temac, ganmtkicdeba didi 

saxazavis skalis (cxadia, am terminis gareSe) codna. 

12.3. patara saxazavebi, xazva da zomva maTi saSualebiT. 

vrwmundebiT, rom patara da did saxazavebze moniSnuli 

santimetrebi erTmaneTis tolia. Semdeg viwyebT xazvas 

rveulis furcelze, patara saxazaviT. dasaxazi monakveTis 

sigrZe Tavidan ar aRemateba 9 sm-s, raki ariTmetikis xaziT 

ufro didi ricxvebis Cawera jer ar gviswavlebia. vaxazvinebT 

mocemuli zomebis mqone kvadratebs, marTkuTxedebs, texilebsa

- 34 - 

da sxva. geometriaSi Semogvaqvs axleburi cxrilebi _ 

gadanomril monakveTTa sigrZeebisa. II.10 

rveulSi daxazuli monakveTebis zomva, ukve 20 sm-is 

farglebSi. milimetrebs pirvel klasSi ar vaswavliT, amitom 

yvela sigrZe santimetrebis mTeli ricxvs Seicavs, raTa ar 

morCes milimetrebiani naSTi. 

mocemuli sigrZis gverdebis mqone marTkuTxedis an 

kvadratis daxazva uredebian furcelze. mocemuli sigrZis 

monakveTTa mqone gaxsnili texilebis daxazva, roca sigrZeebi 

mocemulia cxrilis saxiT. ori mravalkuTxedis gverdebis 

gazomva. Sedegebis Cawera cxrilSi. mravalkuTxedis yvela 

gverdis sigrZeTa jamis gamoTvla. I.6, I.7 

13.3. uadvilesi amocanebi manZilebsa an moZraobaze, romelTa 

amosaxsnelad saWiroa erTi erTeuliT gazomil manZilTa Sekreba 

an gamokleba. maTi amosaxsneli TvalsaCino sqemebi. II.10 

sigrZeTa Cawera matebisa an klebis mixedviT. garda amisa, 

vimeorebT da vajamebT nakvTebis naswavl dajgufebebs _ mTw- 

Si mocemuli sqema-suraTis mixedviT. moZraobisas warmoqmnil 

sivrciT da droiT mimarTebaTa erTdrouli dakavSireba 

naswavl rigobiT ricxvebTan. Tanmimdevroba sivrcesa da droSi. 

14.3. SedarebiT rTuli, ormoqmedebiani amocanebi moZraobaze. 

mocemuli sigrZis texilis daxazva. logikuri sqemebi 

mimarTebaze da kerZo / zogadi. ZiriTadi naswavli sakiTxebis 

gameoreba, ganmtkiceba da ganviTareba _ nusxa ix. $ 5-Si. I.5 

§ 8. sagakveTilo monakveTebi da gakveTilebis ageba 

Cven vgulisxmobT, rom gakveTilis xangrZliobaa 40 wuTi 

(an 45). Cveni yoveli gakveTili isea dagegmili, rom moicavs 

5-7-wuTian TamaSs, xatvas, gaferadebasa Tu sxva saxaliso 

moqmedebebs. amitom aqtiuri Sesveneba TviT gakveTilis 

ganmavlobaSive xdeba. 

sagakveTilo monakveTebi gadanomrilia or-ori ricxviT. 

magaliTad, me-20 gakveTilis monakveTebia: 20.1, 20.2, 20.3, 

20.4. maTgan pirveli ricxvi, 20, gviCvenebs gakveTilis nomers.

- 35 - 

meore ricxvi, 1, 2 an 3, gviCvenebs imas, rom am sagakveTilo 

monakveTSi muSavdeba sakiTxi 1-li, me-2 an me-3 xazidan. xolo 

20.4 _ esaa maTematikuri TamaSi. yoveli sagakveTilo 

monakveTisTvis meore ricxvi 4 _ TamaSis aRmniSvnelia. 

sxvadasxva gakveTilze sxvadasxva TamaSi tardeba _ gakveTilis 

romelime sakiTxTan dakavSirebiT. 

zogjer, Tuki aucilebelia, masw.-s SeuZlia Secvalos sagakveTilo 

monakveTTa Tanmimdevroba, magaliTad, me-20 gakveTilze 

jer daamuSaos monakveTi 20.3, Semdeg ganixilos saSinao 

davaleba (19.5, 19.6 da sxva), Semdeg daamuSaos monakveTi 20.1, 

Semdeg moawyos TamaSi 20.4, Semdeg daamuSaos monakveTi 20.2. 

magram sanam masw.-i sagakveTilo monakveTebis Tanmimdevrobis 

Secvlas gadawyvets, winaswar unda daukvirdes, isini 

erTmaneTze xom araa damokidebuli, magaliTad, mesameSi iseTi 

am xom araa naxsenebi, rac pirvelSi an meoreSi unda 

gveswavlebina. xolo TamaSis monakveTi ki sasurvelia, rom 

yovelTvis SuaSi iyos, meore monakveTis Semdeg, mesamemde. 

TamaSis monakveTi moswavlis rveulSi TiTqmis arsad araa, 

radganac am TamaSebs ar sWirdeba rveulSi muSaoba, isini 

zepirad tardeba. 

zogjer moswavlis rveulSi gamotovebulia agreTve sxva 

monakveTic: meore nomriT 1, 2 an 3. esec imas niSnavs, rom 

Sesabamis safexurs ar sWirdeba rveulSi muSaoba. igi zepirad, 

dafasTan an raime sxva saxiT tardeba. xolo gegma-konspeqtebSi 

ki arcerTi sagakveTilo monakveTi araa gamotovebuli, maT 

Soris arc TamaSebisa. 

garda dasaxelebuli oTxi monakveTisa, gakveTili moicavs 

ganxilvas saSinao davalebisa, romelic klass yoveldRiurad 

eZleva. Tavidan mxolod Zalian cota, erTi-ori gasaferadebel-dasaxatavi 

davaleba, Semdeg sxvagvarebic. TandaTan saSinao 

davalebaTa raodenoba matulobs oTx-xuTamde. 

saSinao davalebis amocanebi gadanomrilia ise, rom maTi 

meore nomeri iwyeba 5-dan. magaliTad, me-19 gakveTilis saSinao 

davalebis amocanebis nomrebia: 19.5, 19.6, 19.7, 19.8, 19.9.

- 36 - 

maTgan pirveli davaleba (nomriT 19.5) _ saweri an weris 

winare savarjiSoa, romelic ujredebian badeSi sruldeba. 

zogjer amgvari savarjiSo klasSive iwyeba da masw.-i amowmebs, 

Tu ramdenad sworia moswavleTa xelis moZraobani werisas. 

Semdeg moswavleebi Sin amTavareben nimuSis mixedviT weras. 

klasma droulad unda CainiSnos saSinao davaleba. masw.-i gadaSlis 

msr-s saWiro adgilas, miuTiTebs klass saTanado adgils 

(TiTiT, radganac bavSvebs didi ricxvebi ar ecodinebaT). 

moswavleebi male mieCvevian saSinao davalebis advilad da 

swrafad CaniSvnas (rveulebSi SemoxazviT). aseve unda moxdes 

xolme saxelmZRvanelos an rveulis saWiro gverdze gadafurcvla 

da saWiro adgilis moZebna saklaso muSaobis drosac. 

xatvisa an xelgarjilobis gakveTilze moswavleebs unda 

davamzadebinoT geometriuli naWrelebiT galamazebuli grZeli 

saniSnebi. Semdeg moswavles msr-Si yovelTvis unda edos 

saniSni, raTa swrafad gadaSalos xolme igi saWiro adgilas. 

uamisod gakveTilze bevri dro daikargeba fuWad! 

davalebas moswavleebi imave msr-Si asruleben. 

zogjer klass evaleba raimes daxatva. masw.-i gaafrTxilebs 

bavSvebs, rom daxaton ise, rogorc SeZleben, advilad da 

swrafad. mimsgavseba araa saWiro, naxati SeiZleba sqematuri 

iyos. es mSoblebmac unda gaiTvaliswinon! 

davalebebi 20.5, 20.6 da ase Semdeg _ ganixileba momdevno, 

21-e gakveTilze _ sasurvelia, rom es gakveTilis dasawyisSive 

moxdes. xolo me-20 gakveTilis dasawyisSi ganixileba 

davalebebi 19.5, 19.6 da ase Semdeg. 

rogorc viciT, saswavlo wlis ganmavlobaSi ramdenjerme 

masw.-s SeuZlia SeaCeros programaSi winsvla _ raime Zneli 

sakiTxis gasameoreblad da gansamtkiceblad. SesaZloa, SeCerebis 

mizezi moswavleTa mier gakveTilebis gacdenebis didi dagrovebac 

iyos. amitom gakveTilis namdvili kalendaruli rigiTi 

nomeri acdeba (gauswrebs) mis nomers gegma-konspeqtebSi. 

magaliTad, SeiZleba moxdes, rom gegma-konspeqtebis 67-e gakve- 

Tili gaviaroT saswavlo wlis ara 67-e, aramed 69-e an 70-e

- 37 - 

gakveTilze. Tumca, gakveTilebis numerciisas Cven amas ugulebelvyofT 

da gakveTilebs vnomravT miyolebiT. es masw.-s 

araviTar sirTules ar Seuqmnis _ is xom mainc gegmakonspeqtebs 

unda mihyves, nomers ra mniSvneloba aqvs! 

unda damkvidrdes wesi _ arcerTi gakveTili saSinao 

davalebis gareSe. es unda icodes moswavlemac da mSobelmac. 

im gakveTilebze, romlebzec masw.-i SeaCerebs dagegmil 

programaSi winsvlas, saSinao davaleba Tavad unda SeurCios 

klass ganvlili masalidan. amisaTvis masw.-ma yovelTvis unda 

moniSnos xolme is amocanebi, romelTa gavla Tavis droze raime 

mizezis gamo ver moeswro. SeCerebisas pirvel rigSi es 

amocanebi unda misces klass saSinao davalebad. garda amisa, 

winaswar unda SearCios erTxel ukve micemuli iseTi amocanebi, 

romelTa mcireodeni, araarsebiTi Secvla advilia da isinic 

misces klass saSinao davalebad. xolo roca gadavlen meore 

msr-Si, SeCerebisas saSinao davalebad SeiZleba mieceT 

ukanaskneli nawilidan SerCeuli iseTi amocanebi, romlebsac 

ar sWirdeba saklaso damuSaveba (amgvari amocanebi mravladaa). 

msr-s is ukanaskneli amocanebi, romelTa gavla veRar 

moeswreba saswavlo wlis ganmavlobaSi, klass miecema sazafxulo 

saardadegebo davalebad. 

amrigad, magaliTad, me-20 gakveTilze muSavdeba pirveli 

xazis me-20 sagakveTilo monakveTi (misi nomeria 20.1), meore 

xazis me-20 sagakveTilo monakveTi (misi nomeria 20.2) da mesame 

xazis me-20 sagakveTilo monakveTi (misi nomeria 20.3). Tvi- 

Teuls eTmoba saSualod 7-15 wuTi. darCenili 10-15 wuTidan 

naxevarze meti eTmoba wina, me-19 gakveTilze micemuli saSinao 

davalebis Semowmebasa da sajaro ganxilvas (saswavlo wlis 

dasawyisSi es nawili araa), naxevarze naklebi ki _ maTematikur 

TamaSs (misi nomeria 20.4). ase aigeba yvela gakveTili. 

ufro xSirad (Tumca ara yovelTvis) yvelaze meti dro 

sWirdeba pirveli xazis sakiTxebs. amitom gakveTilze maT 

eTmoba SedarebiT meti dro _ 10-15 wuTi (rogorc moiTxovs). 

Sesabamisad, meore da mesame xazis sakiTxebs naklebi dro

- 38 - 

eTmoba xolme _ 5-10 wuTi. 

masw.-ma amas sagangebo yuradReba unda miaqcios. amisaTvis man 

xSir-xSirad unda daixedos saaTze. ise ar unda moxdes, rom 

pirvelma da meore sagakveTilo monakveTebma STanTqos mTeli 

gakveTilis dro da veRar moeswros verc mesame sagakveTilo 

monakveTi da verc TamaSi. 

erTsadaimave sakiTxze 10-15 wuTze meti xniT SeCerebas 

azri ara aqvs. pirvelklasels ar ZaluZs amdeni xniT erT 

sakiTxze fiqri. Tundac klass gauWirdes raime sakiTxi, misi 

ganxilva mainc droulad unda momTavrdes, radgan meti xniT 

masze SeCereba _ drois fuWad gaflangva iqneba. am sakiTxs 

masw.-i unda moubrundes meore dRes. Tuki saWiro iqneba, man 

programaSi winsvlac unda SeaCeros erTi dRiT (aq ar 

viTvaliswinebT sust, ganviTarebaSi CamorCenil moswavleebs, 

romelTac damatebiTi sagangebo muSaoba sWirdebaT, gakveTilebis 

Semdeg an uqme dReebSi). 

sazogadod, SeiZleba moxdes, rom gakveTilze ver moeswros 

saTanado masalis damuSaveba. am SemTxvevaSi morCenili nawili 

damuSavdeba momdevno gakveTilze. Tuki es morCenili nawili 

moiTxovs 5 wuTze met dros, maSin axali sakiTxebi aRar unda 

damuSavdes, samagierod, moxdes naswavlis gameoreba. saSinao 

davalebad ki klass mieces Sesabamisi saSinao davalebis 

daaxloebiT naxevari. 

rogorc iTqva, sagakveTilo monakveTTa Tanmimdevroba araa 

arsebiTi. mTavaria is, rom TviTeuli maTgani safuZvlianad damuSavdes. 

safuZvlianadve unda gairCes wina gakveTilze micemuli 

saSinao davalebac. moswavleebma kargad unda gaiazron 

misi TviTeuli amocanis amoxsna, agreTve sakuTari Secdomebi 

Tu xarvezebi. swored am dros xdeba namdvili, aqtiuri 

swavleba: roca moswavles ukve winaswar damoukideblad aqvs 

nafiqri sakiTxze da Semdeg klasSi es sakiTxi sajarod 

aqtiurad gairCeva. 

saSinao davalebis erToblivi ganxilvisas masw.-i Camoivlis 

merxebs Soris da amowmebs, Tu rogor Seasrules moswavleebma

- 39 - 

esa Tu is amocana. Tan ramdenime moswavle rig-rigobiT 

gamoTqvams Tavis azrs. erTerTi maTgani dafasTanac SeiZleba 

iqnes gaZaxebuli, raTa Tavisi amoxsna aCvenos. bolos masw.-i 

kiTxulobs: _ vinmes xom ara aqvs sxvagvari pasuxi? _ da Tuki 

aseTi aRmoCndeba, isic gairCeva. 

moswavleebs am dros kalmistrebi Tu fanqrebi ukaviaT. Tuki 

zogma maTganma ver amoxsna esa Tu is amocana, an bolomde 

ver daasrula, iqve Seavsebs Tavis xarvezebs. masw.-i sul 

merxebs Soris dadis, raTa yovelive amas Tvalyuri adevnos. 

romeli amocanis amoxsnasac ar sWirdeba msjeloba, 

azrovneba da romelsac ara aqvs amoxsnis sxvadasxva SesaZleblobani 

_ masze naxevar an 1 wuTze metxans SeCerebas azri 

ara aqvs. sxvagvar amocanebs ki 2-4 wuTi eTmoba. cxadia, am 

droSi SeuZlebelia yvela moswavlis amoxsnaTa ganxilva, 

masw.-is mier erTi Tvalis gadavleba ki araa sakmarisi. amitom 

masw.-i unda zrunavdes imaze, rom Tuki romelime moswavles 

erT gakveTilze ar mouwia saSinao davalebis Sesaxeb azris 

gamoTqma, Semdeg gakveTilebze mainc mouwios. 

saazrovno amocanaTa umravlesoba iseTia, rom maTi pasuxis 

oradori SesaZleblobaa: marTebuli da mcdari. magram zogi 

amocana ara mxolod saazrovno, aramed agreTve SemoqmedebiTicaa: 

moswavles evaleba, TviTon moigonos axleburi ram 

(magaliTad: raime formebisgan Sedgenili naWrela, an axleburi 

formis rvakuTxedi, an raime sayofacxovrebo amocana `-iT 

naklebobaze~). amgvar amocanebs marTebuli amoxsnis mravali 

SesaZlebloba aqvs. masw.-s uyuradRebod ar unda darCes 

arcerTi moswavlis amoxsna. yvela marTlac axleburi amoxsna 

sajarod unda waikiTxos an dafaze daxazos, xolo moswavle 

Seaqos. saWiroa wamaxalisebeli niSnis dawerac. Semoqmedebas 

gansakuTrebuli waxaliseba sWirdeba! 

SedarebiT Zneli amocanebi * niSniTaa moniSnuli. moswavleebsac 

da mSoblebsac ecodinebaT, rom maTi amoxsna araa 

yvelasaTvis savaldebulo. 

winaswarve unda hqondes wakiTxuli meTod-saxelmZRva-

- 40 - 

neloTa yvela paragrafi (garda am saxelmZRvanelos momdevno 

nawilisa). gansakuTrebiT saguldagulod unda hqondes Seswavlili 

maTematikur-meTodikuri sakiTxebi. xolo Semdeg, gakve- 

Tilebis Catarebisas, mas dasWirdeba am saxelmZRvanelos ukanaskneli 

nawili _ gegma-konspeqtebi. magaliTad, me-20 gakveTilisaTvis 

masw.-i waikiTxavs Semdeg monakveTebs: 19.5, 19.6, 19.7, 

19.8, (SesaZloa, agreTve 19.9), 20.1, 20.2, 20.3 da 20.4. 

gakveTilebis gegma-konspeqtebi 

codna mxolod maSin aris namdvili codna, roca 

igi sakuTari azris daZabviTaa SeZenili da ara 

marto damaxsovrebiT. l. tolstoi 

gegma-konspeqtebis wakiTxvas ara aqvs azri, Tuki masw.-s winaswar 

kargad ar waukiTxavs zms. uamisod masw.-s ar ecodineba 

swavlebis Cveneuli meTodika, maTematikuri ganmartebani, mravali 

meTadikuri miTiTeba da sxva. masw.-i ver gaiazrebs, ra ras 

ukavSirdeba, ratomaa es ase da ara ise da sxva. mas mouwevs 

miTiTebaTa uazrod, meqanikurad Sesruleba _ rac dauSvebelia. 

Cveni zogadi meTodikis mixedviT, gakveTilze mTavaria kargad 

dasmul SekiTxvaTa Tanwyoba. mxolod amgvarad, aqtiurad, 

Seiswavleba sakiTxi. amitom gegma-konspeqtebSi swored saWiro 

SekiTxvaTa nimuSebia mocemuli. magaliTad: _ saidan viciT, 

rom es raodenoba ufro metia, vidre es? 

gegma-konspeqtSi mocemulia mxolod SekiTxva, isic mSralad 

da mokled, magram igulisxmeba, rom masw.-i mas saTanadod 

ganavrcobs, magaliTad: _ aba, gela, miTxari, saidan viciT, rom 

es raodenoba ufro metia, vidre es? garda amisa, masw.-is mier 

kargi gamoTqmiT dasmul SekiTxvas unda axldes saTanado 

mimika da xelis moZraobebic. magaliTad, sityvaze ès~ masw.-ma 

TiTi unda daados saTanado grovas. garda amisa, xelebis moZraobiT 

unda gaaTvalsaCinoos sxvadasxva sityvebi, magaliTad: 

`xazi~, `Zalian bevri~, `Semcirda~, `SevkriboT, SevaerToT~ da 

sxva. es yovelive Zalian mniSvnelovania. 

SekiTxvebis savaraudo da mosalodneli pasuxebi mocemulia

- 41 - 

figurul frCxilebSi. magaliTad: _ saidan viciT, rom es 

raodenoba ufro metia, vidre es? {dawyvilebis Semdeg aq 

morCa ai es rgoli}. Tuki pirveli moswavle, romelsac SekiTxva 

davusviT, kargad ver gvipasuxebs, vekiTxebiT sxvebs. 

sxva moswavlis kargi pasuxi pirvelsac unda gavameorebinoT. 

Semdeg unda vkiTxoT kidev erTs (magaliTad, SedarebiT pasiur 

an sust moswavles), oRond cota sxvagvarad, magaliTad, ase: 

_ manana, axla Sen mipasuxe, am grovebidan romelSia 

naklebi raodenobis rgolebi? ratom? 

Tuki Tavidanve vercerTma moswavlem ver gagvca kargi 

pasuxi, saWiroa misaxvedri miTiTebiT daxmareba, magaliTad: _ 

am grovebis ra gavakeTeT Cven? {grovebis dawyvileba} 

hoda, ai, SexedeT, aq ra mogvrCa? {Savi rgoli} ese igi, 

romel grovaSi yofila ufro bevri rgoli? 

es yvelaferi unda gaakeTos masw.-ma: SekiTxvis ganvrcoba, 

saTanado gamoTqmis SerCeva, gamosakiTx moswavleTa kargad 

SerCeva, SekiTxvis formis Secvla da klasSi misi `datrialeba~, 

pirisaxis gamometyvelebis Secvla Tu xelebis 

moZraoba. saxelmZRvaneloSi ki mSralad mocemulia mxolod 

erTi SekiTxva, viTarc ZiriTadi nimuSi: _ saidan viciT, rom 

es raodenoba ufro metia, vidre es? saxelmZRvaneloSi 

ufro metis dawera xom SeuZlebelia. 

masw.-s araviTar SemTxvevaSi ar unda daaviwydes, rom saWiroa 

SekiTxvis amgvarad `datrialeba~. mxolod erTi SekiTxva 

da erTi pasuxi araa sakmarisi! gansakuTrebiT didxans unda 

ìtrialos~ axal an Znel sakiTxTan dakavSirebulma SekiTxvam. 

rogorc dasabuTebulia zms $ 1 da $ 12-Si, Cveneuli 

meTodikisTvis miuRebelia moswavleTa gunduri pasuxebi. 

garda amisa, pasuxi, rogorc wesi, moklea da am dros 

moswavle fexze nu adgeba. 

kidev erTi friad mniSvnelovani ram. moswavlis Secdomebi 

da xarvezebi, rogorc wesi, misma Tanaklaselebma unda 

Seasworon Tu Seavson _ da ara masw.-ma. Cven arsad ar vwerT, 

magram yvelgan vgulisxmobT, rom roca moswavles raimes

- 42 - 

pasuxi SeeSleba an dafasTan gamosuli moswavle mTlad kargad 

ver Seasrulebs daxazvisa Tu grovebis dawyobis davalebas, 

masw.-i ikiTxavs: _ aba daukvirdiT, rame xom ar SeSlia 

irines? rogor unda eTqva? (rogor unda daexaza?) 

Sesworebisa Tu Sevsebis Semdeg Tavad irinemac unda gaimeoros 

marTebulad! 

maSasadame, yovel saswavlo sakiTxTan dakavSirebiT mocemuli 

gvaqvs SekiTxvaTa (mSrali da mokle) nimuSebis 

mTliani Tanwyoba. ese igi, masw.-ma ar unda gamotovos arcerTi 

am SekiTxvaTagani _ mas romc miaCndes, rom romelime SekiTxva 

zedmetia. arc SekiTxvis arsebiTi Secvlaa sasurveli. 

sazogadod, rogorc iTqva zemore, masw.-i unda moeridos 

meTodikur xazebSi TviTnebur cvlilebaTa Setanas. 

garda amisa, ar mivuTiTebT kidev erT rames. roca klass 

miecema saweri, dasaxazi, dasaxati, dasaTvleli Tu sxva 

saklaso samuSao, masw.-ma merxebs Soris unda iaros da da 

Tvalyuri adevnos muSaobas: ramdenad kargad asruleben davalebas, 

rogor ukaviaT kalmistari Tu rogor xazaven, vinmes 

rame xom ar eSleba, welSi xom ar ixrebian da sxva. 

masw.-s ar unda daaviwydes merxebs Soris siaruli! 

sazogadod, gegma-konspeqtebze Tvalis gadavlebac ki sakmarisia, 

raTa mkiTxveli darwmundes, Tu ramdenad yovlismomcvelia 

da Tanmimdevruli aqtiuri mzaobis meTodikis ganxorcieleba. 

Zalian iSviaTia masw.-is mier warmoTqmuli TxrobiTi winadadebani, 

isinic TiTqmis yovelTvis mas Semdeg warmoiTqmis, rac Tavad 

moswavleebi ityvian. SesaZloa, arazustad, arasrulad, 

mouxeSavad _ magram mainc ityvian ise, rom cxadi iqneba _ maT 

gaiges. masw.-i mxlod amis Semdeg ityvis TxrobiT winadadebas, 

romelic, arsebiTad, mxolod gaimeorebs, daadasturebs da 

daxvews moswavleTa mier naTqvams. da esec _ Zalian xanmoklea, 

sul erTi-ori mokle winadadeba. masw.-is Tundac erTwuTiani 

monologi, sakiTxis àxsna~ Tu `ganvrcoba~ _ gamoricxulia. 

meTodika isea agebuli, rom, Tuki masw.-i saguldagulod 

Seasrulebs gegma-konspeqtebis miTiTebebs, kerZod, zemore

- 43 - 

aRwerili wesiT daamuSavebs yvela im ljUywjU da cs[bofcjU 

winadadebas, romlebic gegma-konspeqtebis udides nawils Seadgens, 

moswavle TavisTavad iswavlis. Tanac, iswavlis namdvilad, 

anu gaRrmavebulad, aqtiuri codnis gaSinaganebiT. 

gegma-konspeqtebSi saSinao davalebis garCevas mivuTiTebT 

mxolod maSin, roca davalebis romelime amocanas raime 

sagangebo yuradRebis miqceva Tu msjeloba sWirdeba. sxva 

SemTxvevebSi amas ar mivuTiTebT, magram yvelgan vgulisxmobT! 

rogorc wesi, gakveTili iwyeba saSinao davalebaTa erToblivi 

ganxilviT: moswavleebi rig-rigobiT pasuxoben, Tan masw.-i 

merxebs Soris dadis da amowmebs, rogor aqvT Sesrulebuli 

esa Tu is amocana. moswavleebs fanqrebi ukaviaT, raTa 

Seasworon, Seavson da daxvewon TavianTi namuSevari. 

meTodikur miTiTebaTa ZiriTadi nawili mocemulia 

gakveTilebis mixedviT. magram maTi nawili saerToa ramdenime 

sagakveTilo monakveTisaTvis. amgvari miTiTebani mocemulia 

calke qvesaTauriT da calke numeraciiT, saTanado adgilas. 

isini ase aRiniSneba: 

I-3 (I anu ariTmetikis xazis me-3 saerTo meTodikuri 

miTiTeba); II-14 (II anu algebra-gamoyenebiTis xazis me-14 

miTiTeba); III-6 (III anu geometriis xazis me-6 miTiTeba). maTi 

teqstebi sxvagvari SriftiTacaa gamorCeuli. 

saxelmZRvaneloSi mocemuli 1.1, 1.2, 1.3, 1.4 da 1.5 

sagakveTilo monakveTebi da saSinao davalebebi 1.5, 1.6... 

gankuTvnilia ara mxolod pirveli, aramed pirveli da meore 

(SesaZloa, agreTve mesame) gakveTilebisaTvis. es imitom, rom 

Tavidan yvelafrisTvis SedarebiT bevri dro daixarjeba. 

Semdeg ki Cveulebriv gagrZeldeba 2.1-dan. 

Tavidan moswavleebs gauWirdebaT xatva da xazva, albaT arc 

ferTa saxelwodebani ecodinebaT kargad. cxadia, naxatisa Tu 

naxazis xarisxs ara aqvs mniSvneloba. garda amisa, roca moswavle 

ityvis, rom ar icis, Tu rogor daxatos, masw.-i daamSvidebs: eg 

araferi, xatva arc me vici kargad, magram rogorc SevZleb, ise

- 44 - 

davxatav (da dafaze aCvenebs martiv naxats). sazogadod, masw.-ma 

aCvenos xolme klass naxatebis SesaZlo nimuSebi. moswavleebi male 

daeuflebian ferTa saxelwodebebs, fanqris xmarebasac gaeCvevian da 

amis mere didi xalisiT Seasruleben xatva-xazvis davalebebs! 

sazogadod, yvelaze Znelia pirveli erTi Tve, gansakuTrebiT 

_ pirveli erTi kvira. moswavleebs ara aqvT TiTqmis 

araviTari unarCvevebi, ver ikaveben fanqars, ver igeben naTqvams, ver 

ikreben yuradRebas... masw.-i amas nu SeuSindeba, man unda icodes, 

rom Tuki keTilsindisierad Seasrulebs Cveneul miTiTebebs 

TandaTan yvelaferi mogvardeba da kalapotSi Cadgeba. xolo 

saswavlo wlis bolos iseTi Sedegebi eqneba, rasac albaT 

TviTonac ar moeloda! 

iwyeba gakveTili 1, rasac mokled aRvniSnavT ase: g 1/ 

g 1/!cfwsj!/!dpub/!ejej!/!qbubsb/!xsf!eb!njtj!ebybuwb/ 

pirvel gakveTilze masw.-i klass aCvenebs, Tu rogor unda 

gadafurclon msr an mTw miTiTebul adgilas, gaSlili wigni 

rogor unda daidon merxze, rogor unda moZebnon saWiro 

adgili. auxsnis, rom msr _ rveulia da masSi unda xaton, xazon, 

weron. TandaTan misi ydac unda gaaferadon. xolo mTw- 

Si araferi ar unda daiweros, igi sufTad unda Seinaxon. 

Semdeg klass daavalebs, rom Semdegi gakveTilidan skolaSi 

moitanon da yoveldRiurad Tan ataron saswavlo nivTebi: 

kalmistari; feradi fanqrebi an flomasterebi; saSleli; 

saxazavi wriuli, kvadratuli, samkuTxa da sxva xvrelebiT. 

cxadia, TviTeul am nivTs aCvenebs (saWiroa amis Tqma 

mSoblebisTvisac). 

Semdeg Catardeba xuTwuTiani TamaSi: magnitur dafaze 

areulad daewyoba sul sxvadasxva zomis wreebi, aTiode (an 

Cveulebriv dafaze feradad daixateba, an lursmnebian dafaze 

Camoikideba). oRond sxvadasxva zomis wreebs feric sxvadasxva 

unda hqondes, an zogi Wreli unda iyos. 

_ axla daviwyoT TamaSi-Sejibri, vnaxoT, vin gaimarjvebs.

- 45 - 

oRond icodeT, vinc wamoiZaxebs, imisi pasuxi ar CaiTvleba da 

is waagebs! miTxariT, rogor unda vTqvaT pasuxi? {xelis 

aweviT} magram vinc xels tyuilad aiwevs da pasuxi SeeSleba, 

mas mere aRar vaTqmevineb sxva pasuxs. amitom visac gamarjveba 

unda, jer kargad moifiqros da xeli mere awios. viwyebT! aba, 

yvelaze swrafad vin ityvis, ra ferisaa yvelaze didi wre? 

Semdeg masw.-i moacilebs yvelaze did wres da ityvis: _ 

ganvagrZoT TamaSi! aba, yvelaze swrafad vin ityvis, ra ferisaa 

darCenilebSi yvelaze didi wre? 

ase ganmeoredeba ramdenjerme. gamovlindeba ramdenime gamarjvebuli. 

morCenili dro daeTmoba sagakveTilo monakveTebs _ ramdenic 

moeswreba. danarCenebi _ Semdegi gakveTilisTvis gadava. 

pirvel gakveTilze klass saSinao davaleba ar eZleva. 

---------------------------------------------------------------------------------- 

momdevno teqsti gansxvavebuli SriftiTaa dabeWdili da mas 

numeraciac gansxvavebuli aqvs. es imis maCvenebelia, rom es 

teqsti exeba mTel Tavs _ da ara mxolod momdevno gakveTils. 

J.2/!ebxzwjmfcb/!nfu.oblmfcpcbtUbo!eblbwTjsfcvmj!fopcsjwj! 

tbljUyj/ am TavSi mTavari sakiTxia dawyvileba: ori grovis wevrTa 

erTmaneTTan saTiTaod Sewyvileba (erTi aqedan, erTic iqidan; erTi 

aqedan, erTic iqidan da ase Semdeg). ori grovis dawyvileba 

Sedgeba nivTTa mravali calkeuli Sewyvilebisagan, magram mainc 

erTianad unda aRviqvaT. amitom nivTebis grovaTa dawyvileba unda 

moxdes erTi adamianis mier, viTarc erTi mTliani moqmedeba! 

sasurvelia, masw.-ma daicvas sizuste: raodenobaze iTqmis `meti / 

naklebi~ (magaliTad, girCebis raodenobaa naklebi, vidre kubikebis 

raodenoba). xolo TviT sagnebze iTqmis `bevri / cota~ an, SeiZleba, 

`meti / naklebi~ (magaliTad, girCebi ufro cotaa, vidre kubikebi, 

girCebi naklebia, vidre kubikebi). sxvaTa Soris, grovebisaTvis 

ufro zustia `mravali / cota~, xolo `bevri~ dauTvlad niv- 

Tierebaze iTqmis, magaliTad, haerze, wyalze, fqvilze, qviSaze da 

sxva; daTvladze ki iTqmis `mravali~. magram jer-jerobiT am 

naklebadcnobili sityviT sakiTxs nu gavarTulebT (Tumca, Tan-

- 46 - 

daTan, es enobrivi sakiTxic gamosasworebeli iqneba). 

amgvarad, unda vTqvaT: `Cxirebi ufro cotaa, vidre kenWebi~, 

anda `Cxirebis raodenoba naklebia, vidre kenWebisa~ (xan ase da xan 

ise). xolo ùfro bevri raodenoba~ an ùfro cota raodenoba~ 

es enobrivi Secdomaa. 

iTqmis agreTve `didi / mcire raodenoba~, magram es gamoTqmebi 

ar unda vixmaroT, `didi / patara~ anu `didi / mcire~ sivrciTi 

mimarTebebisaTvisaa umjobesi. 

----------------------------------------------------------------------------------! 

2/2/! cfwsj! /! dpub/! nlwfUsbe! hbotywbwfcvm! sbpefopcbUb! 

nfupcb-!oblmfcpcb/!zwfmb{f!cfwsj!/!dpub/!!!masw.-s momzadebuli 

aqvs sxvadasxva saTvleli nivTebis grovebi, romlebic raodenobiT 

mkveTradaa gansxvavebuli, magaliTad: 21 wiTeli da 13 

mwvane magniti, 12 kubiki da 19 girCa, 30 Cxiri da 20 kenWi 

(yoveli raodenoba unda iyos 10-ze meti, raTa zogma moswavlem 

daTvla ar daiwyos). TviTeul SemTxvevaSi SekiTxvaa: _ 

wiTlebi ufro bevria Tu mwvaneebi? kubikebi ufro bevria Tu 

girCebi? romelia ufro cota _ Cxirebi Tu kenWebi? 

SekiTxva `ratom~ jer ar daismis. met-nakleboba aSkarad 

Cans. oRond, am gakveTilze raodenobaTa met-nakleboba ar unda 

daupirispirdes nivTebis zomebis sidide-simcires. amitom 

Sesadarebel grovebSi daaxloebiT toli zomebis mqone nivTebi 

unda iyos. araa saWiro, rom, magaliTad, wignebis grova 

SevadarebinoT Cxirebis grovasTan. 

masw.-i zogjer SekiTxvas ase dasvams: _ wiTlebis 

raodenobaa meti Tu mwvaneebisa? oRond, moswavleebs jer nu 

vaTqmevinebT sityvas `raodenoba~ (Tuki TviTon gaimeoreben, 

kargi iqneba). 

Semdeg masw.-i klass aCvenebs oTx grovas, magaliTad: 5 

burTi, 10 girCa, 18 suraTi da 30 kubiki. jer Sedardeba 

burTebis raodenoba girCebisas, girCebisa _ suraTebisas, 

suraTebis _ kubikebisas. bolos daismis SekiTxva: _ amaTgan 

romelia yvelaze bevri? {kubikebia yvelaze bevri} yvelaze 

cota? {yvelaze cotaa burTebi}.

- 47 - 

aseTive SekiTxvebi daismis msr-is suraTi 1.1-is Sesaxebac: 

_ wignebi ufro bevria Tu burTebi? burTebis raodenobaa meti 

Tu fanqrebisa? romelia yvelaze bevri: wignebi, fanqrebi Tu 

burTebi? yvelaze cota? 

TfojTwob; termini `kubiki~ Seesatyviseba saTamaSo nivTs, 

xolo `kubi~ _ geometriul sxeuls. 

2/3/!ejej!/!qbubsb/!!hbotywbwfcb!!cfwsj!/!dpubtbhbo/!!zwfmb{f! 

ejej! /! qbubsb/! ! ! masw.-i klass aCvenebs did wiTel da patara 

Sav kubiks (an, Sesabamisad, ori feris did-patara burTebs). _ 

romelia ufro didi, wiTeli Tu Savi? Savia ufro patara Tu 

wiTeli? _ SeiZleba Tu ara, rom ase vTqvaT: `wiTeli burTi 

ufro bevria, vidre Savi~? {ara, wiTeli burTi ufro bevri ki 

araa, ara _ ufro didia}. ase rom vTqvaT: `Savi burTi ufro 

cotaa, vidre wiTeli~? {arc es varga!} aba rogor unda 

gveTqva? {ufro pataraa} 

zedsarTavis `patara~ sinonimia `mcire~. magram jer-jerobiT 

am naklebadcnobili sityviT sakiTxs nu gavarTulebT. 

masw.-i dafasTan gamoiyvans xuT moswavles. _ romelia 

amaTgan taniT yvelaze didi? yvelaze patara? SeiZleboda 

Tu ara, rom gveTqva `yvelaze bevri~? `yvelaze cota~? aba 

rogor unda vTqvaT? 

2/5/!Tfnefh!ebwbmfcbt!UbnbTj.Tfkjcsfcjt!tbyf!njfdfnb/!! 

jer masw.-i dafaze daxatavs patara kacunas da carciT 

aCvenebs klass, Tu ras niSnavs naxatis Semoxazva (irgvliv 

Sekruli xaziT). 

ganvixilavT msr-is suraTs 1.4. _ aba, yvelaze Cqara vin 

ipovis yvelaze did kacunas? SemoxazeT igi mwvaned (an 

lurjad)! yvelaze Cqara vin ipovis yvelaze patara kacunas? 

SemoxazeT igic! yvelam ipova? yvelam Semoxaza? aba, vis 

SeeSala? axla darCenil kacunebSi moZebneT swrafad yvelaze 

didi kacuna! SemoxazeT igi wiTlad! vin moaswro? xom ara 

SegSlia? axla darCenilebSi moZebneT yvelaze patara kacuna, 

SemoxazeT wiTlad! 

masw.-i daivlis merxebs Soris da visac gauWirda, yvelas

- 48 - 

aCvenebs, Tu risi gakeTeba moeTxoveboda. 

2/3! `! hbhs[fmfcb/ moswavleebma unda Seavson 1.2 cxrilis 

carieli ujredebi: Sig saTanado zomis sagnebi unda Caxaton. 

terminebi `cxrilis striqoni da sveti~ jer araa saWiro. 

masw.-ma TiTis dadebiT aCvenos saTanado ujredebi da ikiTxos: 

_ ra xatia aq? {Citi} aq? {aqac Citi} aq? {aqac Citi} aq? 

{aqac} romelia yvelaze didi Citi? gaaferadeT yviTlad! 

yvelaze patara Citi romelia? gaaferadeT lurjad! es ujredi 

ki carielia. aq ra unda exatos? {saxli} ramxela saxli? 

{yvelaze didi saxli} CaxateT! ra unda exatos aq? {yvavili} 

ramxela yvavili? {yvelaze patara yvavili} CaxateT! danarCeni 

ki Sin CaxateT! saxlSi SeavsebT danarCen cariel ujredebs. 

aba, SemoxazeT es nomeri! (mivuTiTebT 1.2/1.6-ze) es geqnebaT 

saSinao davalebad. masw.-i dafaze dawers am nomers da 

aCvenebs, ras niSnavs misi Semoxazva. 

---------------------------------------------------------------------------------- 

JJJ.2/! [jsjUbej! oblwUfcjt! dopcb/! oblwUjt! vdwmfmpcb! njtj! 

gfsjtb! bo! nefcbsfpcjt! Tfdwmjtbt/! ! geometriul nakvTebTan 

dakavSirebiT mTavaria, rom moswavleebma amTaviTve gaacnobieron: 

nakvTis raoba ar icvleba misi feris Secvlisas an mdebareobis 

Secvlisas iseve rogorc ariTmetikaSi raodenoba ar icvleba 

grovis ganlagebis SecvliT! (es Sedareba mxolod masw.-isTvisaa da 

ara moswavleTaTvis). wrisaTvis es yvelaze advilia, radganac misi 

TvalsaCino `xedi~ ar icvleba motrialebiT. magram, magaliTad, 

kvadratisa Tu marTkuTxedisaTvis sakmaod Zneli gasaazrebelia. 

umTavresi geometriuli nakvTebis ganxilvis Tanmimdevroba 

ganpirobebulia ariTmetikis xaziT: dawyebuli 7.3-dan, ricxvs 

òri~ emTxveva orboloiani monakveTi, ricxvs `sami~ samkuTxedi 

da ase Semdeg. 

moswavleebi swavloben nakvTebis cnobas mTliani formiT. araa 

saWiro amaTuim nakvTis (samkuTxedis, kvadratis, xuTkuTxedisa da 

sxva) gansazRvrebis magvari ram. es jer Zalian naadrevia. arc xuTkuTxedisa 

Tu eqvskuTxedis gverdebisa an wveroebis raodenobis 

daTvlaa Tavidan saWiro. moswavleebma aq ganxiluli, am

- 49 - 

Taviseburi formis xuTkuTxedi Tu eqvskuTxedi unda icnon 

mTliani formiT, erTbaSad, da ara raimes daTvlis Sedegad. 

gverdebis raodenobis sakiTxi ramdenime gakveTilis Semdeg 

damuSavdeba (14.3-Si).!! 

---------------------------------------------------------------------------------- 

2/4/!xsf/!njtj!ebybuwb!)yfmjU*!eb!njtj!ebyb{wb!xsjvmywsf. 

mfcjboj!tbyb{bwjU!)eb!Tfnehpnj!hbgfsbefcjU*/!masw.-i miuTiTebs 

msr-s suraTze 1.3, marcxena wreze. _ vinmem Tu icis, es 

wiTeli ra aris? {es aris wre} axla mwvaned gaaferadeT mis 

gverdiT wre. kidev erTi wrec gaaferadeT Savad. romeli wrea 

yvelaze didi? {wiTeli wre} yvelaze patara? {Savi wre}. 

_ axla wiTeli wris gverdiT kidev davxazoT wreebi. 

moimarjveT saxazavebi, daadeT furcelze ise, rom mrgvali 

xvreli aq moxvdes. axla SemoxazeT! axla Tqven ra feradac 

gindaT, im ferad gaaferadeT! 

Semdeg kidev davaxazvinebeT erT wres, Semdeg kidev erTs. 

masw.-i Tan merxebs Soris ivlis da moswavleebs miexmareba. 

aucilebelia wris Semoxazuli konturis (wrexazis _ arcerTi 

es termini araa saWiro!) gaferadeba! gauferadebeli jer kidev 

araa wre, araa dasrulebuli naxazi! [ix. zms $ 5]. 

2/6/! xfsjt! txbwmfcjt! Tftbyfc! {phbej! njUjUfcboj! jy/! qbsb. 

hsbgTj! 5/! ! masw.-i klass jer daanaxvebs, Tu rogor unda 

daikavon kalmistari. cxadia, amas bavSvebi ase male ver 

aiTviseben. masw.-i auxsnis, rom moswavles evaleba, badeSi 

kalmistriT daxazos iseTive xazebi, rogorebic mocemulia 

rveulSi. moswavleebi jer mkrTalad daxazul xazebze 

gadaatareben kalmistris wvers, Semdeg TviTon daxazaven. 

amas sul ramdenime wuTi eTmoba _ ramdenic saWiroa yvela 

xazis dasaxazad. 

TfojTwob; SemdgomSi weris sagakveTilo monakveTi gegmakonspeqtSi 

ganxiluli aRar iqneba _ yvelaferi aseve unda 

gameordes. mxolod saweri nimuSebi TandaTan garTuldeba. 

Tanac, rogorc iTqva paragrafSi 4, klass es savarjiSoebi 

ufro xSirad saSinao davalebad eZleva.

- 50 - 

2/8/! cpmpt! lmbtt! ljefw! njfdfnb! tbTjobp! ebwbmfcb; nomrebi 

1.7 da 1.8 (igive suraTebi meore gakveTilisTvisacaa saWiro da 

msr-Si amitom aqvs maT ormagi nomrebi). masw.-i klass auxsnis, 

rom saWiroa mxolod gaferadeba, meti araferi. 

---------------------------------------------------------------------------------- 

1.8-Si unda gaferaddes mxolod rgoli, misi Siga xvreli 

ki carieli unda darCes _ rogorc iqvea mocemuli. 

amieridan Cven aRar mivuTiTebT, magram yvelgan vgulisxmobT, 

rom yoveli gakveTilis bolos masw.-ma klass saSinao davalebis 

nomrebi unda Semoaxazvinos. Tavidan bavSvebs es gauWirdebaT, 

magram male mieCvevian da cota xanSi, advilad Semoxazaven xolme. 

rogorc iTqva zemore, saSinao davalebaTa nomrebia 5-dan 9mde, 

magaliTad: 4.5, 4.6, 4.7, 4.8 da 4.9 (an ufro naklebia). 

maSasadame, masw.-s unda axsovdes: yoveli gakveTili iwyeba 

saSinao davalebis erToblivi ganxilviT da mTavrdeba momdevno 

saSinao davalebis micemiT. 

---------------------------------------------------------------------------------- 

g 2. hspwbUb!Tfebsfcb/!ejej!eb!cfwsj/!shpmj-!njtj!ebybuwb/ 

3/1/ tbTjobp!ebwbmfcjt!Tfnpxnfcb. gakveTili, iseve rogorc 

yvela sxva, iwyeba saSinao davalebis ganxilviT. jer-jerobiT 

saSinao davalebani advilia, gansaxilveli araferia, masw.-i 

swrafad Camoivlis merxebs Soris da naxavs, rogor gaaferades 

moswavleebma suraTebi 1.7, 1.8 da rogor daasrules 1.6. 

---------------------------------------------------------------------------------- 

TfojTwob/ SemdgomSi saSinao davalebis ganxilvas mivuTiTebT 

mxolod maSin, roca marTla rame iqneba gansaxilveli da samsjelo. 

sxva SemTxvevebSi arafers mivuTiTebT. 

SemdgomSi, roca saSinao davalebani moicavs namdvil amocanebsac, 

drois dazogvis mizniT Tavidan aRar Semowmdeba is davalebani, 

romlebsac aqvs momdevno gakveTilis nomeric. rogorc viciT, 

amgvari ormagi nomeri imis aRmniSvnelia, rom es amocana sagakveTilo 

monakveTSi isedac unda gairCes, axal sakiTxTan dakavSirebiT. 

amitom misi calke Semowmeba aRar iqneba xolme saWiro. 

----------------------------------------------------------------------------------

- 51 - 

3/2/! sbpefopcjU! peobw! hbotywbwfcvm! hspwbUb! Tfebsfcb! `! 

np[sbwj! hspwfcjt! ebxzwjmfcb! nfupcjt! hbtbslwfwbe/! jer 

gameorebaa: iseve, rogorc 1.1-Si, msr-s 2.1 suraTze erTmaneTs 

Seadareben Citebs, xeebsa da buCqebs. oRond, es ufro swrafad 

unda gakeTdes, vidre wina gakveTilze. Semdeg masw.-i klass 

Seadarebinebs iseTsave grovebs, rogorebic iyo 1.1-Si, oRond 

erTi gansxvavebiT: raodenobani mcired gansxvavdeba, magaliTad: 

24 wiTeli wre da 26 mwvane wre (kargia swored wreebi). 

Tanac, raki grovebi uwesrigodaa, maTi Sedareba Znelia. klasSi 

bunebrivad warmoiqmneba azrTa sxvadasxvaoba: zogi ityvis, rom 

mwvaneebis raodenobaa meti, zogi _ rom wiTlebia ufro bevri. 

_ rogor gavigoT, romelia ufro meti? 

kamaTs mkveTrad vwyvetT da jer am SekiTxvas Riad 

vtovebT. masw.-i klass sxva rames eubneba: _ moicadeT, axla 

sxva rame unda giamboT! erTxel Cems ezoSi bavSvebi 

TamaSobdnen da dawyvildnen. erTma gogonam erT biWunas mokida 

xeli, maT ukan sxva gogonam sxva biWunas mokida xeli da ase 

sxvebmac (masw.-i dafaze daxatavs sqematurad gogonas da 

biWunas erT wyvils, mere meore wyvils). aba, gamodiT, ia da 

daTo, dawyvildiT! ai, esaa bavSvebis wyvili. maSin Cems ezoSi 

ase dawyvildnen bavSvebi (masw.-i Tan dafaze ganagrZobs 

sqematuri wyvilebis xatvas). magram bolos darCa erTi biWuna, 

misi mewyvile gogona aRar iyo (dafaze daixateba kentad 

darCenili biWunac). aba vin moisazrebs, vin metyvis, Cems ezoSi 

gogonebi ufro bevrni yofilan Tu biWunebi? (Tan dafaze 

miuTiTebs). {biWunebi ufro bevrni yofilan} saidan viciT? {is 

erTi biWuna xom meti darCa}. 

Semdeg kvlav grovebs mivubrundebiT. _ iqneb axla vinmem 

moisazros, rogor gavigoT, romelia ufro bevri _ wiTlebi 

Tu mwvaneebi?! ra unda gavakeToT amisaTvis? 

Tuki veravin moisazrebs, masw.-i miaxvedrebs: _ kargi, 

warmovidginoT, rom es mwveneebi viTom gogonebi arian, 

wiTlebi ki _ biWunebi. rogor gavigoT, romelia ufro bevri? 

{davawyviloT} aba gamovides erekle da daawyvilos! {aiRebs

- 52 - 

erT wiTels, gadasvams calke, mas miusvams erT mwvanes da ase 

Semdeg} romeli yofila meti? {wiTlebi} ratom? {imitom, rom 

dawyvilebis Semdeg mwvane wreebi mogvrCa uwyvilod _ maT 

raodenobas ara aqvs mniSvneloba}. 

gakeTdeba kidev erTi amgvari savarjiSo. mas sxva moswavle 

Seasrulebs. 

3/3/! ejejt! ebqjsjtqjsfcb! cfwsUbo/! ! qbubsbt! ebqjsjtqjsfcb! 

dpubtUbo/! ! msr-s 2.2-ze SevadarebT wiTel wreebsa da rux 

wreebs. _ romlebia ufro didebi? {aSkarad Cans, rom ruxebi 

ufro didebia} romlebi ufro patarebia _ wiTlebi Tu 

ruxebi? {wiTlebi} axla SevadaroT raodenoba! ruxebis 

raodenoba ufro metia Tu wiTlebisa? romelia ufro bevri? 

{wiTlebi} ufro cota? {ruxebi} ese igi, wiTlebi ufro 

patarebia, magram isini ufro bevria! xolo ruxebi ufro 

cotaa, magram isini ufro... ufro ra? {ufro didebia!}. 

aseve Sedardeba ukve ganxilul 2.1-ze xeebi buCqebs. {xeebi 

ufro didebia, magram ufro cotaa; buCqebi ufro patarebia, 

magram ufro bevria}. 

Tan masw.-i damatebiT davalebasac miscems klass: _ aba, 

moifiqreT iseTi rameebi, rom erTi ufro didi iyos, vidre 

meore. oRond imas nu gaimeorebT, rac aq vTqviT! Tqven sxva 

rame unda moigonoT! (ramdenime moswavle ityvis Tavis mier 

mogonil magaliTs). 

3/4/! shpmj/! njtj! ebybuwb! )yfmjU*! eb! njtj! ebyb{wb! xsjvm. 

ywsfmfcjboj!tbyb{bwjU!)eb!Tfnehpnj!hbgfsbefcjU*/! 

_ Tu icis vinmem, ra aris rgoli? aba, am nivTebs Soris 

romelia rgoli? (masw.-i klass aCvenebs aTiode nivTs: 

lursmans, WanWiks, Cxirs, asanTis kolofs, ramdenime 

sxvadasxva wres, kubiks da maT Soris erT rgols _ igi 

SeiZleba iyos rkinis, Tunuqis, muyaos...) axla Tqvens 

rveulebSi moZebneT rgolebi! diax! rogor davxazoT rgoli? 

iqve cariel adgilas (1.8. 2.3) moswavleebi or rgols 

daxazaven imave saxazaviT, romliTac wris daxazva iswavles. _ 

jer davxazoT patara wre, Semdeg davadoT saxazavi da

- 53 - 

SemovaxazoT didi wre ise, rom patara wre SuaSi moxvdes. 

cxadia, daxazul rgolebs gaaferadeben, ra feradac surT. 

Semdeg unda gaaferadon 2.3-is qveda nawili. _ aq xatia 

ori saTamaSo kokola (es araa `piramida~!). erTi awyobilia, 

meore ki daSlilia. xom ginaxavT saTamaSo kokola, feradferadi 

rgolebisgan Sedgeba. axla Tqven gaaferadeT orive 

kokolas rgolebi, oRond orive kokolas yvelaze didi 

rgolebi orive gaaferadeT yviTlad! (jer ipovon da 

gaaferadon). axla darCenilebs Soris moZebneT yvelaze didebi 

da isini gaaferadeT wiTlad! 

ase gaaferadeben orive kokolas yvela rgols. ferebis 

gameoreba SeiZleba, oRond ise, rom erTnairi feris rgolebi 

mezoblad, miyolebiT ar aRmoCndes. bolos, kenweroebsac erT 

ferad gaaferadeben. 

3/5/! `nbubsfcmpcbob~. masw.-i gamoiyvans aTiode an naklebi 

raodenobis gogonas. isini erT mwkrivSi erTmaneTs CaeWidebian 

da erT `matarebels~ aawyoben. aseve, masw.-i gamoiyvans maTze 

erTiT naklebi raodenobis biWunebs. isinic `matarebels~ 

aawyoben. Tan ivlian `Cuqu-Cuqu~-Ti. Semdeg gaarkveven, romel 

`matarebelSi~ ufro meti vagonia. amisaTvis erTmaneTis 

gverdiT dadgebian ise, rom gogonebi da biWunebi dawyvildnen. 

_ Turme gogonebs ufro meti vagonebi hqoniaT! 

3/7!!da!!3/8 _ ubralod gasaferadeblebia. 

g 3/!sbpefo/!upmpcb/!tjwsdjUj!njnbsUfcboj/!lvcjt!yfefcj/ 

4/2/! `jnefojwf~/! sbpefopcbUb! upmpcb/! masw.-i winaswar 

dafaze, dafa-magidasa Tu sxvaganac moamzadebs moZravi nivTebis 

Sesadarebel or-or grovas (magaliTad, 17-iansa da 15-iansa, 16iansa 

da 15-ians). gamoiZaxebs ramdenime saSualo moswavles, 

romlebic erTdroulad Seadareben Tav-TavianT grovebs 

dawyvilebiT _ iseve, rogorc 2.1-Si. 

klasic dainaxavs dawvilebis Sedegebs da darwmundeba, Tu 

romeli grovebia meti. 

amasobaSi masw.-i dafaze moamzadebs or tol (12-12-ian)

- 54 - 

grovas da dasawyvileblad gamoiZaxebs kidev erT moswavles. 

dawyvilebis Semdeg aRmoCndeba, rom arcerTi grovidan araferi 

ar morCa. _ romel grovaSia meti raodenobis wreebi? amaSia 

meti? {ara, araferi morCenila} iqneb amaSia meti? {ara} aba 

romelSia meti? {arcerTSi!} diax, arcerTSi araa meti. am 

grovebSi wreebis raodenoba tolia! tolia. arc esaa meti da 

arc es. erTmaneTis tolia. 

es dawyvilebuli grovebi dafaze rCeba. ganvixilavT 3.1 

suraTs. _ ras vxedavT am suraTze? {bavSvebi ezoSi arian} ra 

ukaviaT bavSvebs xelSi? {CanTebi} daakvirdiT, yvela bavSvs 

aqvs Tu ara CanTa? {ki, yvelas aqvs} ramden-ramdeni CanTa 

ukaviaT? {TiTo bavSvs TiTo CanTa, erTs _ erTi} arian Tu ara 

bavSvebi da CanTebi dawyvilebuli? {ki arian _ erTi is da 

erTi is} romelTa raodenobaa meti _ bavSvebisa Tu CanTebisa? 

{tolia} diax, tolia. CanTebi imdenivea, ramdenic _ bavSvebi. 

imdenive. imdenive. (gavameorebinebT erTi-or moswavles). 

masw.-i magitur dafaze `dominoseburad~ daawyobs eqvs 

magnits. gamoiyvans moswavles da daavalebs: _ Senc daawyve 

amdenive magniti! (daawyobs). romelia meti? {imdenivea, toli 

raodenobaa}. 

_ axla vin gviCvenebs, rom Cvens xelebze TiTebis 

raodenoba tolia? {TiTebis dawyvilebiT anu SetyupebiT: ceri 

cers da ase Semdeg}. 

_ gaSaleT orive xeli, daixedeT frCxilebze! aba, TiTebi 

ufro meti gvaqvs Tu frCxilebi? {maTi raodenoba tolia, 

frCxilebi imdenivea, ramdenic _ TiTebi} saidan viciT? 

{frCxilebi da TiTebi dawyvilebulia da arcerTi araa 

morCenili}. 

_ aba, axla moifiqreT iseTi rameebi, rom maTi raodenoba 

toli iyos. oRond imas nu gaimeorebT, rac aq vTqviT! Tqven 

sxva rame unda moigonoT! (ramdenime moswavle ityvis Tavis 

mier mogonil magaliTs. masw.-i Seaqebs da daamatebs: _ diax, 

ciram kargad moifiqra, tansacmelze imdenive Rilia, ramdenic 

_ saRile; naTelac marTalia _ adamians imdenive yuri aqvs,

- 55 - 

ramdenic _ Tvali). 

---------------------------------------------------------------------------------- 

JJ.2/! tjwsdjUj! njnbsUfcboj/! ! mTavari sivrciTi mimarTebani 

zemoT / qvemoT da win / ukan Zalian advilia sakuTari Tavis 

mimarT. magram sxvis mimarT, roca es upirispirdeba sakuTar TavTan 

mimarTebas, cota Znelia. magaliTad, moswavlis ukan dgas skami da 

magida, orive ukanaa, magram skami magidis winaa. am SemTxvevaSi skami 

ùkanaa~ Tavad moswavlis mimarT, magram `winaa~ magidis mimarT. 

cxadia, sazogadod, sivrciTi mimarTebani fardobiTia (...ufro 

maRalia, vidre es, magram ufro dabalia, vidre is...). pirvelklasels, 

misi bunebrivi egocentrizmis gamo, uWirs sxvadasxva 

TvalTaxedvaze dadgoma da yvelafers mxolod sakuTari Tavis 

mimarT aRiqvams. es exeba ara mxolod sivrciT mimarTebebs (magali- 

Tad, logikur da gansakuTrebiT zneobriv mimarTebebSi egocentrizmis 

daZleva mozrdilTa umravlesobasac ki uWirs). 

amitom sivrciTi mimarTebis swavlebasas jer amgvari 

SekiTxvebia: `ra aris amis zemoT (win, ukan da sxva)?~ da momdevno 

gakveTilze ki ufro Zneli: ès ris ukanaa (zemoTaa, winaa...)?~ 

sazogadod, sivrciT mimarTebebSi kargad garkveva logikuri 

azrovnebis ganviTarebis safuZvelia! ix. zms $ 9 

---------------------------------------------------------------------------------- 

4/3/!{fnpU!/!rwfnpU!`!`sb!bsjt!bnjt!{fnpU!/!rwfnpU@~!zwfmb. 

{f!{fnpU!/!rwfnpU/!!jer gavimeorebT mimarTebas didi / patara: 

suraTze 3.1 movaZebninebT yvelaze did bavSvs; yvelaze patara 

bavSvs; yvelaze did CanTas; yvelaze did xes. 

_ axla SexedeT am fanjris rafas. ra aris mis zemoT? 

{farda} ra aris im suraTis qvemoT? {sinaTlis CamrTveli}. 

TfojTwob; aq yvela SekiTxvaze sruli pasuxi unda vaTqmevinoT, 

oRond saxelmZRvaneloSi mokle pasuxebia. 

Semdeg ganvixilavT suraTs 3.2. _ ras xedavT am suraTze? 

{biWuna xis totze zis} biWunas zemoT ra aris? {biWunas 

zemoT mercxalia} mercxlis qvemoT ra aris? {xe} moZebneT xis 

qveda toti! mas ra abia? {msxali} msxali ufro zemoTaa Tu 

biWunas fexebi? {fexebi} biWunas Tavi ufro qvemoTaa Tu xis

- 56 - 

kenwero? {Tavi} xis qvemoT ra aris? {Wa} Waze ra zis? {kata} 

kata ufro zemoTaa Tu msxali? {msxali} ra aris yvelaze 

zemoT? {mercxali} axla yuradRebiT daakvirdiT _ yvelaze 

qvemoT ra aris? {xvliki} biWunasa da msxals Soris? {toti} 

Wasa da tots Soris? {msxali}. 

rveulebi daxuron da daaxloebiT amdagvarsave SekiTxvebs 

upasuxon warmodgeniT. bolos, TviTon moifiqron imis axali 

magaliTebi, rom rame rameze zemoT an qvemoT iyos. 

4/4/! lvcj! eb! njtj! tywbebtywb! yfefcj/! ! masw.-i klass aCvenebs 

xis did kubs (umjobesia, es ar iyos saTamaSo kubiki). _ ra 

hqvia amas? {kubi}. 

aCvenebs saTamaSo kubiks. _ es Tu aris kubi? {ki aris}. 

aCvenebs sxva, aguris formis mqone nivTs: _ es Tu aris 

kubi? {ar aris} ratom ar aris? {imitom, rom wagrZelebulia}. 

yvela Cvenebisas masw.-i sxeulebs abrunebs sxvadasxva 

kuTxeebiT, raTa moswavleebma maTi sxvadasxva xedebi dainaxon. 

Semdeg aCvenebs òrmag kubiks~ _ ori erTnairi kubiki rom 

erTmaneTzea Sewebebuli da ise gaferadebuli, rom erT 

mTlian sxeuls, paralelepipeds qmnis. oRond aCvenebs 

mxolod winxeds _ kvadrats. _ es Tu aris kubi? 

mosalodnelia, rom klass esec kubi egoneba. magram maswavlebeli 

Semoabrunebs sxva mxriv. _ aris es kubi? {ara, ar 

yofila!} winadan ki kubi gegonaT, magram Turme ar yofila! 

Semdeg aCvenebs kidev erT kubs, oRond ise, rom klasi 

xedavs mis mxolod winxeds. _ es Tu aris kubi? {ar viciT, ar 

Cans mTlianad}. SemovabrunebT da davrwmundebiT, rom kubia. 

vekiTxebiT magidaze dawyobil kubebze: 

_ romelia amaTgan yvelaze didi kubi? {mwvane} yvelaze 

patara? {Savi} axla romeli darCa yvelaze didi? {wiTeli} 

axla? {lurji} axla romelia yvelaze patara? {TeTri}. 

magidaze isev vabrunebT yvela kubs. _ gamovides SoTa da 

es kubebi ise daawyos, rom yvelaze didi kubi iyos yvelaze 

qvemoT, yvelaze patara kubi ki _ wiTeli kubis qvemoT iyos. 

(SoTa daawyobs) dajeqi. ra feris kubia mwvane kubis zemoT?

- 57 - 

yviTelis qvemoT? (da sxva) 

Semdeg sul cota xniT vaTvalierebT msr-Si suraTs 3.3. _ 

ai, ase ixateba kubi. suraTze kubi mTlianad Cans Tu ara? 

{ara} suraTze mTlianad ver gamovaCenT. wre rom davxazoT, Tu 

gamoCndeba mTlianad? {ki, wre gamoCndeba} rgoli? {rgolic 

mTlianad gamoCndeba} kubs ki ver gamovaCenT mTlianad. 

4/5/!`txsbgj!qbtvyfcj~/ masw.-i rig-rigobiT svams SekiTxvebs 

zemoT / qvemoT mimarTebis Sesaxeb: romelia ufro qvemoT _ 

piri Tu cxviri? Tvalis zemoT ra aris? saxlis saxuravis 

qvemoT ra aris? Tma ufro zemoT gvaqvs Tu yuri? manqanis 

borbali ufro zemoTaa Tu misive kari? (sityva `misive~ 

gamokveTilad warmoiTqmis). 

4/7/ da 4/8 _ mxolod gasaferadebelia. 

g 4/!ebxzwjmfcb-!spdb!fsUj!v[sbwjb-!{fnpU/rwfnpU-!lwbesbuj/ 

5/2/! ebxzwjmfcb-! spdb! fsUj! hspwb! v[sbwjb/! gasameoreblad 

ganvixilavT suraTs 4.1. gairkveva, romelTa raodenobaa naklebi 

(amjerad naklebiT daviwyoT) da ratom _ burTebisa Tu 

kubebisa. amasobaSi masw.-i magnitur dafaze daxatavs 15 

rgols, erTmaneTisagan Sori-Sors. iqve miakravs 14 moZrav 

wres, maTac gafantulad. _ SevadaroT es grovebi. rgolebia 

meti Tu wreebi? rogor gavigoT? {davawyviloT} magram 

rgolebi daxatulia, maT rom ver aviRebT? ra vqnaT? 

{rgolebs nu aviRebT, ise mivusvaT TiTo wre} gamovides 

arCili da daawyvilos. romlis raodenoba yofila ufro meti? 

(rgolebisa) saidan viciT? ... 

_ axla gavarkvioT, Cven saklaso oTaxSi moswavleebi 

ufro metia Tu merxebi. rogor ggoniaT? (Catardeba 

dawyvileba, risTvisac TiTo merxze TiTo moswavle dajdeba). 

bolos, dawyvildeba da Sedardeba magnitur dafaze 

`dominoseburad~ ganlagebuli xuTi magniti da aseve 

`dominoseburad~ daxatuli eqvsi wre. 

5/3/! {fnpU! /! rwfnpU! `! `ft! sjt! {fnpUbb! /! rwfnpUbb@~! jer 

gasameoreblad ganvixilavT kvlav 3.2 suraTs, oRond

- 58 - 

mexsierebiT, warmodgeniT (visac gauWirdeba, msr gadaSalos). 

SekiTxvebi igivea, rac 3.1-Si, oRond emateba amgvarebi: _ 

msxali ris qvemoTaa? ris zemoTaa? (amgvari wyobis SekiTxvebi 

ufro Znelia). 

Semdeg moimarjveben fanqrebs da 4.2 CarCoSi Caxataven masw.is 

karnaxiT: sul zemoT patara mze; mzis qvemoT patara 

Rrubeli; Rrublis qvemoT patara Citi; sul qvemoT mwvane 

balaxiani miwa; miwis zemoT patara yvavili; yvavilis zemoT, 

oRond Citis qvemoT saxli... masw.-ma dafaze areuli ganlagebiT 

daxatos sqematuri, gaadvilebuli naxatebi _ nimuSebad. Tan 

masw.-i ambobs: _ xateT swrafad, rogorc geadvilebaT. 

5/4/!lwbesbuj!`!lvcjt!dbmj!xbyobhj!)bn!ufsnjojt!hbsfTf*/! 

masw.-i daawyobs: did kubs, masze òrmagkubparale-lepipeds~ 

da masze patara kubs. gairkveva, romeli aris kubi da romeli 

ar aris kubi. _ amis zemoT ra devs? {patara lurji kubi} 

amis qvemoT ra devs? {didi wiTeli kubi}. 

Semdeg masw.-i klass aCvenebs kubs, mis mxolod winxeds, 

TiTiT Semoxazavs mis erT waxnags (es sityva araa saWiro) da 

ityvis: _ ai amas ewodeba kvadrati. kvadrati. 

Semdeg aCvenebs amave kubis sxva waxnags, sxva kubis waxnags, 

aguredis waxnags da gairkveva, maTgan romeli aris kvadrati 

da romeli _ ara. _ es ratom ar aris kvadrati? {imitom, rom 

wagrZelebulia}. 

am gakveTilze yvela waxnags vaCvenebT mxolod sworad, 

cerad gadaxris gareSe! 

msr-s suraTi 4.3. _ ra aris aq daxazuli? {kvadratebi} 

gaaferadeT! yvelaze didi kvadrati gaaferadeT yviTlad; 

yvelaze patara kvadrati _ Savad; sxvebi _ sxvadasxva ferebad, 

rogorc gsurT. 

5/5/! `hvebTj! yfmjt! Dbzpgb! eb! hbnpdopcb~/ masw.-i naWris 

gudaSi Cayris kubebs, aguredebs, piramidebs, burTebsa da sxva 

sxeulebs. moswavleebma rig-rigobiT unda Cayon gudaSi xeli, 

aiRon erTi sxeuli da gudaSi Cauxedavad, xeliT mosinjviT 

swrafad gamoicnon, es sxeuli kubia Tu ara. pasuxis Semdeg

- 59 - 

moswavle amoiRebs sxeuls da yvela naxavs, SeeSala Tu ara. 

Semdeg sxeuls isev Caabruneben gudaSi, masw.-i aurevs da 

momdevno moswavles Caayofinebs xels. 

5/7!eb!5/8!_ mxolod gasaferadebelia/! 

g 5.!ebxzw/!yb{fcjU/!nbSbmj/ebcbmj/!lwbes/!tywbebtywb!nefc/! 

6/2/!ebxzwjmfcb-!spdb!psjwf!hspwb!v[sbwjb!`!yb{fcjU!Tffs. 

Ufcjt! h{jU/! ! masw.-i winaswar moamzadebs dafaze daxazul 14 

rgolsa da iqve mikrul amdenive moZrav kvadrats. Sedardeba 

rgolebisa da kvadratebis raodenoba. gamova is moswavle, 

romelsac jer arcerTxel ar mouwia sakuTari xeliT 

dawyvilebam. gairkveva, rom raodenobani toli yofila, ese igi, 

kvadratebi imdenivea, ramdenic rgolebi. 

amasobaSi masw.-i Cveulebriv dafaze daxatavs 11 varskvlavsa 

da 10 Tolias, gafantulad, erT grovas ufro marjvniv, 

meores _ marcxniv. klass davavalebT, rom daawyvilon (swored 

daawyvilon, da ara mxolod pirdapir Seadaron erTmaneTs) 

varskvlavebi da Toliebi. _ rogor davawyviloT? naxatebs xom 

ver aviRebT! ra vqnaT? Cven gvinda dawyvileba: erTi varskvlavi 

_ erTi Tolia, erTi varskvlavi _ erTi Tolia da ase Semdeg. 

rogor gavakeToT? 

albaT, romelime moswavle moisazrebs. gamovides da 

daawyvilos. Tuki vercerTma ver moisazra, saWiroa misaxvedri 

miTiTeba: _ aba maSin me getyviT. aviRoT carci da... da... ra 

vqnaT? {davxazoT xazebi da am xazebiT davawyviloT}. 

Tuki arc aman uSvela, masw.-i TviTon SeaerTebs xaziT (araa sa- 

Wiro, rom xazi swori iyos!) erT wyvils. danarCenebi ki moswavlem 

SeaerTos. _ axla miTxariT, romlis raodenobaa meti? ... 

Semdeg klass davavalebT, rom daawyvilon (swored 

daawyvilon, da ara pirdapir Seadaron erTmaneTs) 5.1 suraTze 

mocemuli grovebi. _ rogor davawyviloT ZaRlebi da lekvebi? 

{xazebiT SevaerToT TiTo ZaRlTan TiTo lekvi}. 

aseve _ xoWoebi da sokoebi, rgolebi da wreebi. 

6/3/!nbSbmj!/!ebcbmj/!hbombhfcb!tjnbSmjt!njyfewjU/!zwfmb{f!

- 60 - 

nbSbmj! /! ebcbmj/! jer ganvixilavT suraTs 5.2. daismis yvela 

SesaZlo SekiTxva zemoT / qvemoT mimarTebaze: _ ra aris 

rgolis qvemoT? ris zemoTaa patara wre? ris zemoTaa didi 

wre? ra aris kubsa da patara wres Soris? ris da ris 

Sorisaa didi kvadrati? ... da ase Semdeg. 

Semdeg davakvirdebT mTw-is ydis suraTs (mTlianad, 

gadaSlilad). SekiTxvebi Seexeba mimarTebas maRali / dabali: 

daTunia ufro maRalia Tu wero? biWuna Tu kata? ku ufro 

dabalia Tu cuga? romeli cxovelia aq yvelaze maRali? 

yvelaze dabali? 

Semdeg TviTon moigonon axali magaliTebi imisa, rom erTi 

rame meoreze dabalia an maRalia. 

bolos, gamoviyvanT moswavleebs, gavarkvevT, romeli 

romelze maRal-dabalia da ras niSnavs `dadgoma simaRlis 

mixedviT~. moswavleebs davsvamT da xelaxla davavalebT: _ 

gamodiT da dadeqiT simaRlis mixedviT! 

Tuki zogi ori moswavle toli simaRlisa aRmoaCndeba, 

erTmaneTis win dgebian, ese igi, rigSi erTsadaimave adgilas. 

6/4/! lwbesbuj! tywbebtywb! nefcbsfpcbTj! `! lvcjt! npusjbmf. 

cjU/!vaCvenebT sxvadasxva sxeulTa waxnagebs (jer gadauxrelad, 

sworad) da vekiTxebiT, aris es kvadrati Tu ara. Semdeg 

viwyebT waxnagTa cerad gadaxras. moswavleebma unda gaiazron, 

rom Tuki kvadrati iyo, kvadratadve darCeba, Tuki ara _ ara. 

_ kvadrati ginda sworad iyos, ai ase, da ginda cerad da 

daxrilad, ai ase, mainc kvadratia! esec kvadratia da esec. 

Semdeg daevalebaT, rom suraTze 5.3 gaaferadon mxolod 

kvadratebi. sxvebi ar unda gaaferadon! 

6/5/ klasi daiyofa or an sam gundad (rigebis mixedviT, an 

gogonebi da biWebi). gundebi Seejibrebian erTmaneTs: masw.-is 

taSis Semokvraze romeli gundi moaswrebs gamosvlas, 

simaRlis mixedviT dadgomas da gaCumebas. dadgoma ar 

CaeTvlebaT, sanam Tundac erTi bavSvi idgeba mcdarad (TviTon 

unda mixvdnen _ romeli), da sanam gundi ar gaisuseba. 

6/7-is piroba cota grZelia. masw.-i auxsnis mas klass.

- 61 - 

g 6.!hbnfpsfcb/!xjo!/!vlbo-!Tpsjt/!!xfsujmj/ 

7/2/!Ubwj!2/2.jt!Tfkbnfcb!eb!hbnfpsfcb/!masw.-i dafaze winaswar 

moamzadebs dasawyvilebel grovebs: moZrav kvadratebsa da 

uZrav rgolebs, moZrav wreebsa da uZrav Toliebs, uZrav 

navebsa da uZrav niCbebs; magidaze _ kubebsa da girCebs. zog 

grovaSi toli raodenobis nivTebi unda iyos. oTxi-xuTi 

moswavle erTad imuSavebs: daawyvileben, Seadareben, daskvnebi 

sajarod gairCeva. 

Semdeg yvela erTad daawyvilebs msr-Si 6.1 suraTis 

rgolebsa da kvadratebs. gairkveva, romlebia meti da ratom. 

7/3/!xjo!/!vlbo/!Tpsjt/!`sb!bsjt!bnjt!xjo!/!vlbo@~!zwfmb{f!xjo! 

/!vlbo/!!_ vin zis xatias win? kokasa da sandros Soris? am 

rigSi yvelaze win? yvelaze ukan? da ase Semdeg. 

Semdeg masw.-i gamoiyvans da erTmaneTis pirispir daayenebs 

or moswavles, vTqvaT, ciras da nikas. _ kari ciras winaa Tu 

ukan? {ciras ukanaa} nikas winaa Tu ukan? {nikas winaa} ratom 

moxda, rom kari nikasTvis winaa, magram cirasTvis ukanaa? 

{imitom, rom erTmaneTis pirispir dganan, rac cirasTvis 

ukanaa, nakasTvis is _ winaa} fanjrebi nikas winaa Tu ukan? 

{nikas ukanaa} ciras winaa Tu ukan? (da ase Semdeg). 

Semdeg 6.2 CarCoSi masw.-is karnaxiT Caxataven: SuaSi xe; xis 

win manqana; xis ukan magida, oRond magida ufro dabali unda 

iyos, vidre manqana; xis zemoT mze; magidis qvemoT burTi; da 

ase Semdeg: muSavdeba naswavli sivrciTi mimarTebani: zemoT / 

qvemoT, maRali / dabali, win / ukan. rogorc yovelTvis, masw.-i 

dafaze daxatavs areulad sqematur naxatebs da axsenebs bavSvebs, 

rom xaton advilad da swrafad, rogorc eadvilebaT. 

Tan moswavleebma moigonon win / ukan mimarTebis axali 

magaliTebi. 

7/4/! xfsujmj! `! lwbesbujt! dbmj! xwfsp! )kfs! bn! ufsnjojt! 

hbsfTf*/!xfsujmjt!tjndjsf/!njtj!ebtnb!gvsdfm{f/!masw.-i kvlav 

aCvenebs klass sxvadasxva sxeulTa waxnagebs xan sworad, xan 

cerad da ekiTxeba, aris es kvadrati Tu ara.

- 62 - 

_ gadaSaleT rveulebi aq (aCvenebs imave 6.2-is gverds). 

daakvirdiT yvelafers, rac aqaa dawerili, daxazuli da 

daxatuli. aba vin mixvdeba, ra aris yvelaferze ufro mcire, 

yvelaferze ufro patara? kargad daaTvaliereT wignis es 

erTi gverdi! ra aris yvelaze patara? {wertili} wertilze 

ufro patara sxva araferia! wertili yvelaze patara ramea! 

Semdeg masw.-i TiTs daadebs didi kubis erTerTi waxnagis, 

kvadratis wveros: _ ai es wveti _ esaa erTi wertili. 

furcelze rogor unda davsvaT wertili? {kalmistris 

wveriT} aba am cariel adgilas dasviT wertilebi! 

masw.-i merxebs Soris ivlis da Tvalyurs adevnebs muSaobas. 

yuradRebas amaxvilebs imaze, rom wertili patara unda iyos. 

xolo Tuki msxvilia _ es aRaraa wertili. es iqneba patara 

Savi wre! 

7/5/! `txsbgj! qbtvyfcj~/ SekiTxvebi exeba naswavl sivrciT 

mimarTebebs zemoT / qvemoT, maRali / dabali, win / ukan. 

magaliTad: cxeni ufro maRalia Tu adamiani? Cveni skolis 

ezoSi WiSkris ukan ra aris? manqanas saWe ufro win aqvs Tu 

skamebi? ... 

7/7/-Si saWiroa dakvirveba _ kanonzomierad gagrZeleba! 

g 7.!psfvmj-!sjdywj!psj/!`sbnefoj~. xjo!/!vlbo/!npoblwfUj/! 

--------------------------------------------------------------------------------! 

J.3/! sjdywfcj! fsUjebo! frwtbnef/! Semdegi gakveTilidan iwyeba 

axali sakiTxebi: ricxvebi erTidan eqvsamde. ricxvebi iswavleba mxolod 

TvalsaCinod mocemuli nivTebis grovebiT, cifrul niSanTa 

da CanawerTa gareSe. ZiriTadi sayrdenia dominosebri TvalsaCinoeba. 

mTavaria imis gaazreba, rom raodenoba ar icvleba grovis ganlagebis 

SecvliTa da misi gafarToebiT. mcire raodenobebisaTvis 

amis danaxva advilia, magram mainc aucileblad unda warmovaCinoT. 

yvela raodenoba erTnairad Semogvaqvs da erTidaimave savarji- 

Soebs viyenebT. asea sagakveTilo monakveTebSi: 7.1, 9.1, 10.1, 11.1, 12.1, 

13.1. magaliTisaTvis ganvixiloT 10.1 ricxvi oTxi. jer Semogvaqvs 

TvalsaCinod mocemuli raodenoba oTxeuli. masw.-s winaswar

- 63 - 

dasawyvileblad momzadebuli aqvs sxvadasxva oTxeulTa oTxi wyvili. 

Sesadarebeli oTxeulebi sxvadasxva ganlagebisa unda iyos: 

* * * * * * * * * * 

* * * * * 

da sxva (zogi uwesrigocaa). * 

vixilavT sxvadasxva oTxeulebs sxvadasxva nivTebisa, sxvadasxva 

nakvTebisa da sxvadasxva ganlagebiT. Tanac, zogi grova unda iyos 

ufro gafantuli, ganlagebiT ufro `didi~ esaa mTavari! 

moswavleebi dawyvilebiT TviTon rwmundebian, rom yvelas 

raodenoba tolia. tolia, magram ramdenia? oTxia. amgvarad, 

oTxeuli esaa sagnebis oTxeuli, anu oTxi cali sagani. 

Sesadarebeli grovebi masw.-ma winaswar unda moamzados. 

erTmaneTs Sedardeba moZravi da moZravi, moZravi da uZravi da 

uZravi da uZravi grovebi. uZravi grovebia: dafaze daxatuli 

Toliebisa, kacunebisa, naZvebisa da sxva (masw.-is surviliT); moZravi 

grovebia: Cxirebisa, girCebisa, kubebisa da sxva saklaso 

TvalsaCinoebidan. magram, amaT garda, aucilebelia geometriuli 

nakvTebi isini, romlebic im droisaTvis ukve naswavlia: wreebi, 

rgolebi, kvadratebi, monakveTebi, samkuTxedebi da sxva (gansakuTrebiT 

ukanasknelad naswavli nakvTebi). maTgan zogi SeiZleba 

daxatuli iyos, zogi ki moZravi (magnituri). 

klasi darwmundeba, rom, miuxedavad sxvadasxva ganlagebisa, 

yvela oTxeulSi oTxi sagania. Semdeg masw.-i dafasTan gamoiyvans 

ramdenime moswavles da daavalebs, daxaton patara rgolebis 

oTxeulebi, TiTo moswavlem TiTo. Tanac, sxvadasxvanairi 

oTxeulebi, sxvisi daxatuli ar gaimeoron. mosalodnelia, rom 

moswavleebi gaimeoreben im sxvadasxva ganlagebas, romlebic masw.-is 

mier iyo momzadebuli. masw.-i miuTiTebs moswavleTa mier daxatul 

oTxeulebze: aqac oTxi rgolia, aqac, aqac, aqac. romelia yvelaze 

lamazi? (gamoiTqmeba Sexedulebani) me mgoni, es jobia yvelas (da 

Semoxazavs `dominosebr~ oTxeuls). 

Semdeg oTxs Sedardeba ukve naswavli ricxvebi: erTi, ori da 

sami. amisaTvis dafaze `dominoseburad~ daewyoba saTanado grovebi 

(oTxeuli da sameuli da ase Semdeg) da dawyvilebiT Sedardeba

- 64 - 

erTmaneTs. 

vimeorebT: masw.-s ar unda aerios: gramatikulad oTxeuli 

arsebiTi saxelia, pasuxobs SekiTxvas `ra?~. xolo oTxi esaa 

raodenoba, ricxvi, romelic pasuxobs SekiTxvas `ramdeni?~. oTxi 

ricxviTi saxelia. cxadia, oTxeuli ufro konkretuli, nivTieri 

da TvalsaCinoa, xolo oTxi ufro ganyenebulia. oTxeuli 

yovelTvis raRacis oTxeulia, xolo oTxi arsebobs TavisTavadac, 

sagnebisagan ganyenebiT. [ix. zms $ 2] 

---------------------------------------------------------------------------------- 

8/2/!psfvmj/!sjdywj!psj/!`sbnefoj~/!fsUj-!psj!Uv!vgsp!nf. 

uj-! bov! nsbwbmj/! masw.-i klass dawyvilebiT adarebinebs 

ramdenime sxvadasxva oreuls. yvelas raodenoba toli 

aRmoCndeba. _ amaTi raodenoba tolia. tolia, magram mainc 

ramdenia? {oria} diax, esec oreulia da esec. amis raodenobac 

oria da amisac, amisac da amisac. 

suraTi 7.1. _ ra aris aq? {ori patara Savi wre} risi 

oreulia aq? {wreebis oreuli} axla Tqven mis gverdiT 

daxazeT lurji wreebis oreuli! kidev dasviT wertilTa 

oreuli! kidev xeliT daxateT patara rgolebis oreuli! 

kidev daxateT patara kacunebis oreuli. 

Tan moswavleebma TviTon moigonon oreulebis axali magaliTebi 

{ori Tvali, ori rqa, velosipedis... ra? ori borbali 

da sxva}. 

sityva `mravali~ damuSavdeba TamaSiT _ ix. 7.4. 

8/3/!xjo!/!vlbo/!Tpsjt/!`ft!sjt!vlbobb!/!xjobb@~! 

jer swrafad gavimeorebT imdagvarsave SekiTxvebs, 

rogorebic iyo 6.2-Si. Semdeg sakiTxi gaZneldeba. 

_ cisana vis win zis? nato vis ukan zis? es skami ris 

ukan dgas? 

Semdeg ganvixilavT mTw-is gaSlil ydas. SekiTxvebi 

imdagvarivea: _ nukris ukan vinaa? cuga vis winaa? kurdReli 

vis ukanaa? es ori pepela vis zemoTaa? ciyvsa da bus Soris 

vinaa? da ase Semdeg. daismis aranakleb 15 SekiTxvisa. 

TfojTwob; suraTze cxovelebi ganpirovnebulia, zRapa-

- 65 - 

riviTaa. amitom daismis SekiTxva `vin?~ (da ara `ra?~). 

8/4/!npoblwfUj!`!lwbesbujt!dbmj!hwfsej )kfs!bn!ufsnjojt! 

hbsfTf*/! npoblwfUj! gvsdfm{f/! npoblwfUjt! psj! cpmp! `! 

xfsujmfcj/!npoblwfUjt!ebyb{wb!tbyb{bwjU/! 

monakveTis, wrfisa da sxva xazebis ganmartebisaTvis ix. zms $ 5.! 

masw.-i klass aCvenebs didi kubis waxnags _ kvadrats. TiTs 

daadebs erT wveros. _ ai esaa erTi... ra? {wertili} es? {esec 

wertilia} eseni _ ori wertilia. erTi wertili, meore 

wertili. ai es ki monakveTia! (TiTs gausvams am ori wertilis 

SemaerTebel wibos _ monakveTs). monakveTi. 

suraTi 7.3. _ moZebneT kvadrati! Savad ra aris Seferadebuli? 

{erTi monakveTi} axla Tqven SeaferadeT kvadratis 

kidev ori sxva monakveTi, erTi wiTlad da erTic mwvaned. 

(unda Seaferadon kvadratis kidev ori gverdi, meoTxis 

Seferadeba araa saWiro). 

Tan masw.-i dafaze daxazavs did monakveTs. _ vin gamova da 

maCvenebs, romeli wertilebia am monakveTis boloebi? diax, 

esenia! ramdeni bolo aqvs am monakveTs? {ori} sxva monakveTi 

rom iyos, mas ramdeni bolo eqneba? {masac ori, yvela 

monakveTs ori bolo aqvs} monakveTis erTi bolo wertilia, 

meore boloc wertilia. 

_ monakveTi rogor unda davxazoT? {saxazaviT} aq daxazeT! 

7.3-is or mkrTal monakveTze gadaaxazaven axal monakveTebs 

da cariel adgilebSi kidev daxazaven or cal monakveTs. yvela 

daxazuli monakveTis bolo wertilebs wiTlad moniSnaven. 

8/5/! `txsbgj! qbtvyfcj~/ masw.-i TviTeul moswavles rigrigobiT 

usaxelebs rames, man ki maSinve unda gamoicnos: erTia, 

oria Tu _ mravali (unda iTqvas mxolod am sami pasuxidan 

erTerTi). Tanac, pasuxi sityviT ki ar unda Tqvas, aramed: 

erTze _ awios erTi TiTi, orze _ ori TiTi, mravalze _ 

gaSlili xelisguli (mravali TiTi). aqve pirvelad Semogvaqvs 

sityva `mravali~. masw.-i ganmartavs, rom es igivea, rac `bevri~. 

am TamaSSi sul ixmareba sityva `mravali~.

- 66 - 

pirvel wreze masw.-i aCvenebs TvalsaCinod: TiTebs, nivTebs, 

suraTze naxatebs... meore wreze ki ukve mxolod sityvierad 

asaxelebs: kiseri, saqanelas Tokebi, qaTmis fexebi, Toxis 

tari, skamis fexebi, adamianis cera TiTebi, spilos fexebi, 

lomis yurebi, xoWos fexebi, xis totebi da ase Semdeg. 

mesame wreze masw.-i TamaSs arTulebs: moswavleTa dasabnevad 

TviTon wevs raodenobis aRmniSvnel TiTebs, magram ganzrax mcdarad. 

magaliTad, ityvis `vaSlis kurkebi~ da awevs or TiTs. 

8/7/!mxolod gasaferadebelia. 

----------------------------------------------------------------------------------! 

TfojTwob/ SemdgomSi amas aRar mivuTiTebT xolme. roca saSinao 

davaleba mxolod gasaferadebelia, an Zalian advilia da arafer 

safrTxilosa da saWoWmanos ar Seicavs, mis Sesaxeb aRarafers aRar 

davwerT. masw.-ma Tavad unda naxos msr-Si mexuTe, meeqvse, meSvide 

(SemdgomSi agreTve merve da mecxre) nomriani davalebani da isini 

misces klass saSinao davalebad. 

---------------------------------------------------------------------------------- 

g 8.!xzwjmj-!psj/!TjhojU/hbsfU/!psj!xfsu/!TffsU/!npoblwfUjU/ 

9/2/!xzwjmj!bov!psfvmj/!psjt!Tfebsfcb!fsUUbo/!_ Cven ukve 

viswavleT dawyvileba. rogor vawyvilebT? ra aris wyvili? 

TandaTan gairkveva, rom wyvili igivea, rac oreuli: _ 

wyvili aris oreuli. sulerTia, romels vityviT. wyvili 

igivea (am sityvis gamokveTiT), rac oreuli. wyvili da 

oreuli erTidaigivea (am sityvis gamokveTiT). 

_ aba moigoneT, ris wyvilebi iciT? {Tvalebis, yurebis, 

xelebis, fexebis, karebis, Zroxis rqebis da sxva} kidev did 

velosipeds aqvs wyvili... (masw.-i dafaze uceb daxatavs 

orTvliani velosipedis sqemas). wyvili ra? {borblebi} kidev 

ra aqvs velosipeds wyvili? ra? {saxelurebi}. 

_ romeli ufro metia, ori Tu erTi? {ori} ratom? 

(davawyvilebinebT oreulsa da erTeuls, oreulSi erTi cali 

morCeba). diax, ori metia, vidre erTi, xolo erTi ufro... ra? 

{erTi ufro naklebia, vidre ori}. 

axla ki Sesruldeba davaleba, romelic samive xazs gaaer-

- 67 - 

Tianebs. masw.-is karnaxiT 8.1 cariel CarCoSi Caxataven: _ 

daxazeT ori wre, erTi zemoT, meore ki _ qvemoT, 

erTmaneTisagan Sori-Sors. zemoTa wreSi, SigniT (am sityvis 

gamokveTiT) daxazeT ori monakveTi da erTi wertili. ori 

monakveTi da erTi wertili! qvemoTa wreSi, SigniT daxazeT 

erTi monakveTi da ori wertili. erTi monakveTi da ori 

wertili! wreebis gareT daxazeT erTi rgoli. rgoli wreebs 

Soris unda iyos! 

9/3/!TjhojU!/!hbsfU/!Ubwj!2/3.jt!hbnfpsfcb!eb!Tfkbnfcb/! 

sivrciTi mimarTeba SigniT / gareT ukve iyo 8.1-Si. 

axla moswavleebi daakvirdebian suraTs 8.2. masw.-is SekiTxvaTa 

meSveobiT suraTebi aRiwereba. Tan saTiTaod gairCeva, 

riT gansxvavdeba zeda suraTi qvedasagan. rac SeiZleba metad 

unda vaTqmevinoT naswavli mimarTebani: didi / patara, zemoT / 

qvemoT, maRali / dabali, win / ukan, SigniT / gareT. Semadarebel 

SekiTxvaTa nimuSebi: _ romeli mdinarea ufro didi 

_ zeda Tu qveda? romeli xidia ufro grZeli? romeli 

saxlis win mdgari xea ufro maRali? romel suraTzea mze 

ufro maRla? Rrubeli ufro dabla? SeadareT Tevzebis 

raodenoba mdinareebSi! didi ixvi zeda suraTzea ufro win Tu 

qvedaze? manqana moswavlis winaa Tu ukan? Robis SigniT 

mdgari xe ufro maRalia Tu Robis gareT mdgari xe? kidev ra 

aris Robis SigniT? kidev ra aris Robis gareT? riT 

gansxvavdeba zeda saxli qveda saxlisagan? mTis ukan risi 

Tavi moCans? diax, esaa eklesiis, taZris gumbaTi. riT gansxvavdeba 

zeda gumbaTi qvedasagan? rogori xeebi moCans Sors, 

ukan? kidev ra gansxvavebas moZebniTT? (da ase Semdeg). 

9/4/! npoblwfUj! tywbebtywb! nefcbsfpcbTj/! psj! xfsujmjt! 

TffsUfcb!npoblwfUjU/!ufsnjoj!`xfsujmfcjt!TffsUfcb~/! 

masw.-i dafaze didi saxazaviT daxazavs erT monakveTs 

sworad da meores _ daxrilad. 

_ ramdeni monakveTia aq daxazuli? gamovides nana da gvaCvenos. 

gamovides goCa da gvaCvenos maTi bolo wertilebi. yvela 

bolo gvaCvenos! ramdeni bolo aqvs erT monakveTs? {ori}.

- 68 - 

masw.-i dafaze dasvams or wertils. _ ramdeni wertilia aq? 

{ori} diax, wertilebis oreulia, wertilebis wyvilia. modi, 

es ori wertili SevaerToT monakveTiT (daadebs did saxazavs 

da SeaerTebs). ra vqeni me? {ori wertili monakveTiT 

SevaerTe} SevaerTe. ra davxaze? {monakveTi} esaa ori 

wertilis SeerTeba (am sityvis gamokveTiT) monakveTiT (Tan 

TiTs gaayolebs im monakveTs). SeerTeba. 

masw.-i kidev dasvams or wertils da aCvenebs, rogor unda 

davadoT saxazavi, raTa isini monakveTiT SevaerToT. 

yuradRebas mivaqcevT, rom saxazavi zed wertilebze ar unda 

idos, odnav qvemoT unda iyos! 

Semdeg moswavleebi msr-Si suraTi 8.3-Si daxazaven monakve- 

Tebs. jer mkrTalad daxazulebze axlebs gadaatareben, mere 

cariel adgilas TviTonac daxazaven. bolos, wertilebis pirvel 

wyvils SeaerTeben monakveTiT, meore wyvilsac SeaerTeben. 

9/5/!`txsbgj!qbtvyfcj~/ SekiTxvebi Seexeba naswavl sivrciT 

mimarTebebs: zemoT / qvemoT, maRali / dabali, win / ukan _ 

mTw-s ydis suraTze. pirvel wreze moswavleebi naxuloben am 

suraTs da ise pasuxoben, meore wreze _ yvela Seinaxavs 

wignebs da warmodgeniT pasuxoben. 

g 9.!tbnfvmj-!tbnj/!trfmj!/!Uyfmj/!tbnlvUyfej!eb!njtj!yb{wb/ 

:/2/! tbnfvmj/! sjdywj! tbnj/! masw.-i klass dawyvilebiT 

erTmaneTs adarebinebs sxvadasxva sameulTa or wyvils (ix. 

saerTo miTiTebani I-2): 

* * * 

dawyvilebiT aRmoCndeba, rom yvelaSi nivTebis raodenoba 

tolia. _ Citebisa da yvavilebis raodenoba tolia. 

monakveTebisa da rgolebis raodenobac tolia. tolia, magram 

mainc ramdenia? {samia} diax, esec sameulia da esec. amis 

raodenobac samia da amisac, amisac da amisac. 


yvela sameulSi sami sagania. Semdeg masw.-i dafasTan gamoiyvans 


* * 

* 

* 

* * 

* 

* 

*

- 69 - 

sameulebi, TiTo moswavlem _ TiTo. Tanac, sxvada-sxvanairi 

sameulebi, sxvisi daxatuli ar gaimeoron. mosalodnelia, rom 

moswavleebi gaimeoreben im sxvadasxva ganlagebas, romlebic 

masw.-is mier iyo momzadebuli. masw.-i miuTiTebs moswavleTa 

mier daxatul sameulebze: _ aqac sami rgolia, aqac, aqac, 

aqac. romelia yvelaze lamazi? (gamoiTqmeba sxvadasxva 

Sexedulebani) me mgoni, es jobia yvelas (da Semoxazavs 

`dominosebr~ sameuls). 

msr-s suraTi 9.1. _ ra aris aq? {sami patara Savi wre} 

risi sameulia aq? {wreebis sameuli} axla Tqven mis gverdiT 

daxazeT lurji wreebis aseTive sameuli! kidev dasviT wertilTa 

sameuli! kidev xeliT daxateT patara rgolebis sameuli. 

_ ras aqvs sami fexi? {zogierT dabal skams, jorkos} ras 

aqvs sami borbali? {sabavSvo samTvlian velosipeds} kidev vin 

moigonebs axal sameuls? 

ricxvi samis amgvari Seswavlis Semdeg masw.-is daxmarebiT 

gadaSlian mTw-s `1-2-3~-ze (cxadia, cifrebi jer araa saWiro 

da maTze yuradRebas ar vamaxvilebT!). _ maCveneT, romelia 

wreebis sameuli. ra feria? romelia wreebis wyvili? ra 

feria? erTi cali wre romaa calke, igi ra feria? 

_ miCvene sami fanqari! daukari taSi samjer! daakakune samjer! 

---------------------------------------------------------------------------------- 

JJ.3/! vgsp! [ofmj! tjwsdjUj! njnbsUfcboj/! gansxvavebiT ukve 

naswavl sivrciT mimarTebaTagan, moswavleebs ver movTxovT 

momdevno sivrciT mimarTebaTa karg dauflebas. maT erTmaneTSi 

aerevaT xolme sqeli da msxvili, viwro da wvrili da sxva 

(samwuxarod, ufrosebsac erevaT xolme). moswavleTa nawils 

SeiZleba gauWirdes agreTve sivrciT mimarTebaTa garkveva 

araTvalsaCinod, warmodgeniT. 

---------------------------------------------------------------------------------- 

:/3/!trfmj!/!Uyfmj!`!UwbmtbDjope/!zwfmb{f!trfmj!/!Uyfmj/! 

masw.-i klass aCvenebs or wigns, romelTagan erTi (vTqvaT, 

Savi) ufro sqelia, vidre meore (vTqvaT, yviTeli). _ ramdeni 

wigni mikavia me? {ori} romelia maTgan ufro sqeli? {Savi

- 70 - 

wigni} ufro Txeli romelia? {yviTeli}. 

aseve Seadareben sqel da Txel mokle ficrebs. Semdeg 

TviTon moswavleebi moigoneben sqeli / Txeli rameebis axal 

magaliTebs. {foToli ufro Txelia, vidre nafoti; TefSi 

ufro sqelia, vidre qaRaldi} masw.-ic dasvams ramdenime amgvar 

SekiTxvas. 

Semdeg ganvixilavT suraTs 9.2. movZebniT yvelaze sqel 

wigns da mas Savad SemovxazavT; yvelaze Txels _ lurjad. 

:/4/!tbnlvUyfej/!tbnlvUyfejt!ebyb{wb/!tbnlvUyfej!tywbebtywb! 

nefcbsfpcbTj/! masw.-i aCvenebs dafaze winaswar daxazul or 

samkuTxeds, romelTagan erTi ufro didia, vidre meore. 

samkuTxedebi unda iyos tolgverda (rogoricaa 9.3 suraTze), 

jer-jerobiT mxolod amgvar samkuTxedebs vixilavT! didi 

samkuTxedi pataras zemoTaa. iqve daxazulia erTi kvadrati da 

sami wre. _ ra aris es? {wre} es? {samkuTxedi} es? 

{kvadrati} es? {samkuTxedi} ramdeni samkuTxedia dafaze 

daxazuli? {ori} kvadrati ramdenia? {erTi} wre ramdenia? 

{sami} zemoTa samkuTxedi ufro didia Tu qvemoTa? {zemoTa} 

qvemoTa samkuTxedi ki ufro... ra? {pataraa} 

suraTi 9.3. _ ra aris es wiTeli nakvTi? (samkuTxedi). 

Semdeg mis gverdiT, Cveulebriv, daxazaven da gaaferadeben 

samkuTxeds. 

saSinao davalebaSic (9.6) daxazaven da gaaferadeben samkuTxedebs. 

damatebiT maxvildeba yuradReba imaze, rom rogorc gvinda 

movatrialoT, samkuTxedi mainc samkuTxedi iqneba. 

:/5/!Tfkjcsj!tjnbSmjt!njyfewjU!ebehpnbTj!`!jy/!6/5/! 

g 10.!pUyfvmj-!pUyj/!gbsUp!/!wjxsp/!lwbesbujt!ebyb{wb/! 

21/2/!pUyfvmj/!sjdywj!pUyj/!njtj!Tfebsfcb!obtxbwm!sjdywfc. 

Ubo/! masw.-i klass dawyvilebiT erTmaneTs adarebinebs 

momzadebul grovebs (ix. saerTo miTiTebani I-2). dawyvilebiT 

aRmoCndeba, rom yvela oTxeulSi nivTebis raodenoba tolia. _ 

kacunebisa da xeebis raodenoba tolia. kacunebi imdenivea, 

ramdenicaa xeebi. samkuTxedebisa da wreebis raodenobac tolia.

- 71 - 

tolia, magram mainc ramdenia? {oTxia} diax, esec oTxeulia da 

esec. kacunebis raodenobac oTxia da xeebisac, amisac da 

amisac. wreebi ramdenia? {oTxi}. 


yvela oTxeulSi oTxi sagania. Semdeg masw.-i dafasTan 

gamoiyvans ramdenime moswavles da daavalebs, daxaton patara 

rgolebis oTxeulebi, TiTo moswavlem _ TiTo. Tanac, sxvadasxvanairi 

oTxeulebi, sxvisi daxatuli ar gaimeoron. 

mosalod-nelia, rom moswavleebi gaimeoreben im sxvadasxva 

ganlagebas, romlebic masw.-is mier iyo momzadebuli. masw.-i 

miuTiTebs moswavleTa mier daxatul oTxeulebze: _ aqac oTxi 

rgolia, aqac, aqac, aqac. romelia yvelaze lamazi? (gamoiTqmeba 

sxvadasxva Sexedulebani) me mgoni, es jobia yvelas (da 

Semoxazavs `dominosebr~ oTxeuls). 

suraTi 10.1.! _ cariel adgilas CaxateT patara wreebis 

aseTi oTxeuli (miuTiTebs `dominosebrze~). gverdiT aq ra 

xatia? Wiqebis ra xatia? {sameuli} lambaqebis? {oTxeuli} 

kidev? {Caidnebis wyvili, oreuli} axla Wiqebis sameuli 

xaziT SeaerTeT qvemoTa sameulTan. ar SegeSaloT _ sameuli 

moZebneT! lambaqebis oTxeuli rasTan unda SevaerToT? {qveda 

oTxeulTan} SeaerTeT! Caidnebis wyvili rasTan SevaerToT? 

{qveda wyvilTan} SeaerTeT! 

_ ramdeni bolo aqvs monakveTs? {ori} axla warmoidgineT 

ori cal-calke monakveTi. sul ramdeni bolo eqnebaT? {oTxi} 

ras aqvs oTxi fexi? {magidas, ZaRls} kidev vin moigonebs axal 

oTxeuls? {manqanis borblebisa, karadis fexebisa da sxva}. 

ricxvi oTxis gacnobis Semdeg masw.-is daxmarebiT bavSvebi 

Slian mTw-s `4~-ze. _ aq, kvadratSi ramdeni wrea? {oTxi} 

aq ki ramdeni wrea? {aqac oTxi} oTxi wre rogor aris aq 

dawyobili? {ori da ori}. axla davTvaloT wreebis raodenoba 

aq! ramdeni wrea? {oTxi} aq rogor aris dawyobili oTxi wre? 

{sami da erTi}. 

moswavleebs wigns davaxurvinebT da mexsierebiT, warmodgeniT 

vapasuxebinebT: _ rogor iyo dalagebuli oTxi wre?

- 72 - 

{ori da ori} kidev rogor iyo? {sami da erTi}. 

Tuki zogi ver gaixsenebs, isev gaSalos wigni da naxos. 

_ miCvene oTxi TiTi! daukari taSi oTxjer! daakakune oTxjer! 

21/3/! gbsUp! /! wjxsp! `! UwbmtbDjope/! ! hbojfsj! bov! gbsUp/! 

zwfmb{f! gbsUp! /! wjxsp/! masw.-i klass aCvenebs toli sigrZis 

or saxazavs, romelTagan erTi ufro farToa, vidre meore. _ 

romelia amaT Soris ufro farTo? {TeTri} TeTri saxazavi 

ufro farToa, vidre wiTeli. wiTeli saxazavi ki ufro... ra? 

{ufro viwroa, vidre TeTri} farTo igivea, rac ganieri. maS, 

am saxazavebidan romeli yofila ufro ganieri? {TeTri} axla 

Tqven moigoneT, rame rom iyos meoreze ufro ganieri. 

Semdeg aseve moigonon viwroze da farToze. 

suraTi 10.2. _ ra xatia aq? {qamrebi} moZebneT ori 

erTnairi qamari da isini mwvaned gaaferadeT. moZebneT yvelaze 

farTo qamari da yviTlad gaaferdeT! moZebneT yvelaze viwro 

da Savad gaaferadeT! 

21/4/! lwbesbujt! ebyb{wb;! lvcjt! ebefcjU! eb! xbyobhjt! Tfnp. 

yb{wjU

- 73 - 

mesameze _ kidev. icvleba TiTebis raodenobac, TviT TiTebic, 

marjvena-marcxena xelic da TiTebis daxrilobac (xan gverdi- 

Taa gaSverili, xan _ qveviT, xan _ zeviT, xan _ cerad ... ). 

g 11.!yvUfvmj-!yvUj/!!ntywjmj!/!xwsjmj/!!yvUlvUyfej/ 

22/2/!yvUfvmj/!sjdywj!yvUj/!njtj!Tfebsfcb!obtxbwm!sjdywfc. 

Ubo/! yvelaferi 10.1-is magvaradaa. masw.-i klass dawyvilebiT 

erTmaneTs adarebinebs momzadebul grovebs (ix. I-2). 

dawyvilebiT aRmoCndeba, rom yvela xuTeulSi nivTebis 

raodenoba tolia. _ navebisa da kenWebis raodenoba tolia. 

navebi imdenivea, ramdenicaa kenWebi. samkuTxedebisa da wreebis 

raodenobac tolia. tolia, magram mainc ramdenia? {xuTia} 

diax, esec xuTeulia da esec. nave-bis raodenobac xuTia, 

kenWebisac. samkuTxedebi ramdenia? {xuTi}. 


yvela xuTeulSi xuTi sagania. Semdeg masw.-i dafasTan 


rgolebis xuTeulebi, TiTo moswavlem _ TiTo. Tanac, 

sxvadasxvanairi xuTeulebi, sxvisi daxatuli ar gaimeoron. 

mosalodnelia, rom moswavleebi gaimeoreben im sxvadasxva 


miuTiTebs moswavleTa mier daxatul xuTeulebze: _ aqac xuTi 


sxvadasxva Sexedulebani) me mgoni, es jobia yvelas (da 

Semoxazavs `dominosebr~ xuTeuls). 

suraTi 11.1. _ aq, cariel adgilas, CaxateT patara wreebis 

aseTi xuTeuli (miuTiTebs `dominosebrze~). gverdiT aq ra 

xatia? kalmistrebis... {oTxeuli} fanqrebis? {sameuli} 

saSlelebis? {xuTeuli} saxazavebis? {wyvili, oreuli} axla 

kalmlistrebis oTxeuli xaziT SeaerTeT qvemoTa oTxeulTan. 

ar SegeSaloT _ oTxeuli moZebneT! 

aseve SevaerTebinebT sxva grovebs. 

_ vin moigonebs raimes sxva xuTeuls? {TiTebis xuTeuli}. 

ricxvi xuTis gacnobis Semdeg masw.-is daxmarebiT bavSvebi 

Slian mTw-s `5~-ze. _ aq ramdeni wrea? {xuTi} aq ki

- 74 - 

ramdeni wrea? {aqac xuTi} xuTi wre rogor aris aq 

dawyobili? {ori da sami}. axla davTvaloT wreebis raodenoba 

aq! ramdeni wrea? {xuTi} aq rogor aris dawyobili xuTi 

wre? {oTxi da erTi}. 


vapasuxebinebT: _ rogor iyo dalagebuli xuTi wre? 

{ori da sami} kidev rogor iyo? {oTxi da erTi}. 


_ miCvene xuTi furceli! daakakune xuTjer! daukari taSi 

xuTjer! 

22/3/! ntywjmj! /! xwsjmj! `! UwbmtbDjope/! zwfmb{f! ntywjmj! /! 

xwsjmj/!!masw.-i klass aCvenebs gaSlil xelisguls, xuT TiTs. 

_ romelia TiTebs Soris yvelaze ufro msxvili? {ceri} ceri 

ufro msxvilia, vidre Sua TiTi. Sua TiTi ki ufro... ra? 

{ufro wvrilia, vidre ceri} romeli TiTia yvelaze wvrili? 

{neki} neki ufro wvrilia, vidre Sua TiTi. Sua TiTi ki 

ufro... ra? {ufro msxvilia, vidre neki} axla Tqven moigoneT, 

rame rom iyos meoreze ufro msxvili. 

Semdeg aseve moigonon wvrilze. Semdeg: _ ra aris Zalian 

wvrili? {Tma, bususi, zogi Zafi}. 

_ monakveTi rogoria _ msxvili Tu wvrili? {wvrilia, 

Zalian wvrilia} monakveTi am yvelaferze ufro wvrilia! 

msr-s suraTi 11.2. _ ra xatia aq? {fanqrebi}. moZebneT is 

fanqrebi, romlebsac erTnairi simsxo aqvs, feri gindac sxva 

iyos, simsxvile hqondeT erTnairi. es ori fanqari SemoxazeT 

Savad! axla moZebneT yvelaze msxvili fanqari! SemoxazeT lurjad! 

axla moZebneT yvelaze wvrili fanqari! SemoxazeT wiTlad! 

22/4/! yvUlvUyfej/! njtj! ebyb{wb! )njbympfcjU*/! axla masw.-i 

pirvelad ixmars termins `nakvTi~. araa saWiro araviTari 

ganmarteba, nurc moswavleebs vaTqmevinebT. sadac aucilebeli 

iqneba, TviTon masw.-i ityvis xolme am sityvas. moswavleebs ki 

vaswavliT SemdgomSi, saTanado dros. 

suraTi 11.3. _ ai es wiTeli nakvTi aris xuTkuTxedi. mis 

gverdiT mkrTalad moniSnulia xazebi da wertilebi. maTze

- 75 - 

daxazeT xuTkuTxedi! gverdiT rom sicarielea, iq Sin kidev 

erTi axali xuTkuTxedi daxazeT. saxazaviT! 

22/5/!`txsbgj!qbtvyfcj~/ masw.-i asaxelebs ricxvs (1, 2, 3, 4 

an 5). moswavlem ki swrafad unda uCvenos imdeni TiTi 

(gamoikveTeba sityva ìmdeni~). 

22/8/! tbTjobp! ebwbmfcfcTj! jxzfcb! obnewjmj! bnpdbofcj/ 

saWiroa ganmarteba, ra moeTxovebaT. isic unda icodnen, rom 

Tuki davaleba daaviwydebaT, Sin vinmes waakiTxon vardisferze 

dawerili teqsti. 

g 12.!frwtfvmj-!frwtj/!njnbsUfcboj!xbsnpehfojU/!frwtlvUyfej/ 

23/1/ masw.-i merxebs Soris CamovliT Seamowmebs, romelime 

moswavles xom ar SeSlia 11.7 amocanis amoxsna. Secdoma 

sajarod gairCeva da Seswordeba. Tan, cxadia, 11.5-is da 11.6-is 

Sesrulebac Semowmdeba. 

23/2/!frwtfvmj/!sjdywj!frwtj/!njtj!Tfebsfcb!obtxbwm!sjdywfc. 

Ubo/! tjuzwb! `Tftbcbnjtj~/! yvelaferi 11.1-is magvarad. masw.-i 

klass dawyvilebiT erTmaneTs adarebinebs momzadebul grovebs 

(ix. I-2). dawyvilebiT aRmoCndeba, rom yvela eqvseulSi 

nivTebis raodenoba tolia. _ isrebisa da qoxebis raodenoba 

tolia. isrebi imdenivea, ramdenicaa qoxebi. xuTkuTxedebisa da 

wreebis raodenobac tolia. tolia, magram mainc ramdenia? 

{eqvsia} diax, esec eqvseulia da esec. isrebis raodenobac 

eqvsia, qoxebisac. xuTkuTxedebi ramdenia? {eqvsi}. 


yvela eqvseulSi eqvsi sagania. Semdeg masw.-i dafasTan 


rgolebis eqvseulebi, TiTo moswavlem _ TiTo. Tanac, 

sxvadasxvanairi eqvseulebi, sxvisi daxatuli ar gaimeoron. 

mosalodnelia, rom moswavleebi gaimeoreben im sxvadasxva 


miuTiTebs moswavleTa mier daxatul eqvseulebze: _ aqac eqvsi 


Sexedulebani) me mgoni, es jobia yvelas (da Semoxazavs

- 76 - 

`dominosebr~ eqvseuls). 

suraTi 12.1.! _ ra aris aq? {eqvsi patara Savi wre} risi 

eqvseulia aq? {wreebis eqvseuli} axla Tqven mis gverdiT 

daxazeT lurji wreebis aseTive eqvseuli! aq kidev xatia 

saTamaSoebis grovebi. jer kubikebis grovaa. igi SeaerTeT mis 

Sesabamis raodenobasTan, qvemoT! 

axla pirvelad gaiJRera sityvam `Sesabamisi~ _ wina gakve- 

Tilebze amgvari davalebebi klass ukve Sesrulebuli aqvs da 

amitom advilad mixvdebian, Tu ra moeTxovebaT. magram mxolod 

es mixvedra araa sakmarisi. saWiroa gamoikveTos sityva 

`Sesabamisi~. amitom mas ramdenjerme gavimeorebT. 

_ yvelam SeaerTa kubikebis grova mis Sesabamis 

raodenobasTan? axla ixuniebis grova SeaerTeT mis Sesabamis 

raodenobasTan. gvanca, Sen Tu SeaerTe SesabamisTan, sxvasTan 

xom ar SegierTebia? aba, vnaxoT! 

da ase gameordeba daTuniebisa da danarCen grovebzec. 

_ ramdeni fexi aqvs xoWos? {eqvsi} vin moigonebs raime 

sxva eqvseuls? {buzis fexebisa} kidev? 

ricxvi eqvsis gacnobis Semdeg masw.-is daxmarebiT bavSvebi 

Slian mTw-s `6~-ze. _ aq ramdeni wrea? {eqvsi} aq ki 

ramdeni wrea? {aqac eqvsi} eqvsi wre rogor aris aq 

dawyobili? {sami da sami}. axla davTvaloT wreebis raodenoba 

aq! ramdeni wrea? {eqvsi} aq rogor aris dawyobili eqvsi 

wre? {oTxi da ori} aq ramdeni wrea? {aqac eqvsi} aq rogorRa 

aris dawyobili eqvsi wre? {xuTi da erTi}. 

moswavleebs wigns davaxurinebT da mexsierebiT, 

warmodgeniT vapasuxebinebT: 

_ rogor iyo dalagebuli eqvsi wre? {sami da sami} kidev 

rogor iyo? {oTxi da ori} kidev? {xuTi da erTi}. 


_ miCvene eqvsi fanqari! daakakune eqvsjer! daukari taSi 

eqvsjer! 

23/3/!obtxbwmj!tjwsdjUj!njnbsUfcboj!`!xbsnpehfojU;!!! 

ejej! /! qbubsb-! nbSbmj! /! ebcbmj-! {fnpU! /! rwfnpU-! xjo! /! vlbo-!

- 77 - 

TjhojU! /! hbsfU-! ntywjmj! /! xwsjmj-! trfmj! /! Uyfmj-! gbsUp! bov! 

hbojfsj!/!wjxsp/!masw.-i svams SekiTxvebs am sivrciT mimarTebaTa 

Sesaxeb. moswavleebi pasuxoben mxolod warmodgeniT, Tvalsa- 

Cinoebis gareSe. 

ZiriTadi SekiTxvebi Seexeba ukanasknel sam wyvil 

mimarTebas, pirveli samis Sesaxeb ki daismeba mxolod TiTo- 

TiTo SekiTxva, gasameoreblad. 

SekiTxvaTa nimuSebi: _ warmoidgine Cveni skolis ezos 

fiWvi da vaSlis xis nergi. romlis tania ufro msxvili? ra 

aris ufro wvrili, vidre nergis tani? 

raze ufro ganieria es zoli? zooparkSi yofilxar? 

bileTebs misi Robis SigniT yidian Tu mis gareT? buStebs? 

Cveni skolis ezos gareT romeli xea? SigniT? 

Semdeg moswavleebi gaakeTeben saxaliso amocanas 12.2. 

Tvali unda gaayolon marcxnidan marjvniv (masw.-i TiTs gaayolebs) 

da rig-rigobiT aRweron sityvierad, Tu ra Seicvala. 

SesaZloa, 12.1-s bevri dro waeRo da amitom 12.2 amocanis 

gakeTeba axla ver moeswros. masw.-ma CainiSnos es amocana da 

klass gaakeTebinos maSin, roca mouwevs winsvlis SeCereba 

erTi gakveTiliT. moswavleebma rac SeiZleba ufro zusti 

gamoTqmebiT unda aRweron TviTeuli cvlileba. Tuki 

gauWirdebaT, masw.-i miexmareba: _ daakvirdi yurebs! daakvirdi 

kuds! (da ase Semdeg). 

23/4/!frwtlvUyfej/!njtj!ebyb{wb!)njbympfcjU*/! 

yvelaferi iseve, rogorc 11.3-Si. 

23/5/! `txsbgj! qbtvyfcj `! tbqjsjtqjspoj~/ masw.-i ambobs 

romelime sityvas naswavl raodenobriv da sivrciT 

mimarTebaTagan: bevri, mravali, cota, meti, naklebi, didi, 

patara, maRali, dabali, zemo, zemoT, zeda, qvemo, qvemoT, 

qveda, wina, win, ukana, ukan, SigniT, Siga, gareT, gare, msxvili, 

wvrili, sqeli, Txeli, farTo, ganieri, viwro. 

moswavlem unda daasaxelos sapirispiro sityva. cneba 

`sapirispiro~ bavSvebma SeiZleba ar icodnen, amitom masw.-i 

Tavidan erTi-or magaliTs etyvis: didi / patara, TeTri / Savi,

- 78 - 

dRe... ? {Rame} ai, esaa sapirispiroebi. 

msxvili, wvrili, sqeli, Txeli, farTo, ganieri, viwro _ 

amaTi sapirispiroebi Znelia. amitom eseni meore wreze unda 

Semovides. Tanac, masw.-i exmareba xelebis moZraobiT: 

xelisgulebiT aCvenebs `sqeli wigni~ an `Txeli wigni~; 

TiTebiT aCvenebs `viwro zoli~ an `farTo zoli~; xelebiT 

aCvenebs `msxvili joxi~ an `wvrili joxi~. 

g 13.!sjdywUb!sjhj/!`tbqjsjtqjsp~. lwbesbuj!tywbebtywb!nefc/ 

23/7-!23/8 da, Sesabamisad, 24/1. _ iseve, rogorc 22/7 da 23/1. 

24/2/! obtxbwm! sjdywUb! Tfebsfcb! fsUnbofUUbo/! sjdywUb! sjhj;! 

fsUj-!psj-!tbnj-!pUyj-!yvUj-!frwtj/!!_ jer gavixsenoT naswavli. 

romelia meti: erTi Tu ori? ori Tu sami? sami Tu erTi? 

kargi. axla gamovides naTia da dafaze daawyos oTxi rgoli 

(gamova da daawyobs dominoseburad). gverdiT daawyve sami 

rgoli! gamovides avTo da aqeT mxares daawyos xuTi rgoli! 

gverdiT daawyve eqvsi rgoli! axla naTia, Sen dawyvilebiT 

gaarkvie, romelia meti: oTxi Tu sami. avTo, Sen ki gaarkvie, 

romelia meti: xuTi Tu eqvsi. 

daawyvileben. _ giorgi, Sexede dafas da miTxari, romeli 

yofila meti: xuTi Tu eqvsi? sruli pasuxi! {eqvsi metia, 

vidre xuTi} oTxi Tu sami? {oTxi metia, vidre sami}. 

Semdeg aseve dawyvilebiT Sedardeba: ori da oTxi; sami da 

xuTi. _ naTela, romeli iqneba meti: eqvsi Tu oTxi? ori 

Tu xuTi? ana, romelia naklebi: erTi Tu sami? ori Tu eqvsi? 

xuTi Tu erTi? 

sanam pasuxoben, masw.-i dafaze erT swor horizontul xazs 

gaavlebs. _ dafasTan gamovidnen nika, cira, gvanca, goCa, zaza 

da lali. nikam daxatos erTi rgoli ai aq (xazis zemoT, 

yvelaze marcxniv), ciram _ ori rgoli ai aq, gvancam _ sami 

ai aq, goCam _ oTxi aq, zazam _ aq xuTi, dalim _ aq (xazis 

zemoT yvelaze marjvniv) eqvsi. rgolebi dasviT ise, rogorc 

am Tqvens wignSia (mTw-Si _ dominoseburad). 

amgvarad, rgolebi erT striqonze unda daixatos. _ axla 

nanam adgilidan waikiTxos (Tan masw.-i TiTs gaayolebs) {erTi,

- 79 - 

ori, sami, oTxi, xuTi, eqvsi} gela, axla Sen miTxari, ratom 

davalageT es ricxvebi ase? ratom sxvanairad ar davalageT? 

ase ratom ar vTqviT: ori, eqvsi, sami, erTi, xuTi, oTxi? 

{imitom, rom jeraa erTi, yvelaze patara ricxvi. merea ori, 

orze didia sami, samze didia oTxi, oTxze didia xuTi, xuTze 

didia eqvsi} amaTgan (es sityva aucilebelia!) romeli ricxvia 

yvelaze didi? {eqvsi} yvelaze patara? {erTi}. 

_ axla aq, cariel CarCoSi (msr-s 13.1), gadaxateT es grovebi! 

Semdeg viwyebT amocanas 13.1. _ aq daxatulia samuSao 

iaraRTa grovebi. ramdeni iaraRia am grovaSi? (vuTiTebT 

pirvel CarCoze) {xuTi} romelia misi Sesabamisi raodenoba 

qvemoT? axla aq waSaleT yvela sxva raodenoba, datoveT 

mxolod xuTeuli! 

danarCenebs Sin Seasruleben (13.6). 

24/3/! obtxbwmj! tjwsdjUj! njnbsUfcboj! xbsnpehfojU/! tjuzwb! 

`tbqjsjtqjsp~/! ! jer gavameorebinebT cnebas `sapirispiro~. _ 

ra aris maRalis sapirispiro? wvrilis sapirispiro? da ase 

Semdeg. Semdeg moswavleebma unda daxaton cariel CarCoSi 

13.2. xataven karnaxiT, swrafad, Tan masw.-i dafaze aCvenebs, advilad 

(geometriul-sqematurad) rogor daxaton esa Tu is 

rame. mTavaria, rom moswavleebma sivrciTi mimarTebani daicvan 

kargad. _ daxateT soflis ori saxli, erTmaneTisgan Sori- 

Sors, erTi maRali, meore _ dabali. orives garSemo Robe dauxateT, 

xazi SemoavleT, viTom Robea. maRali saxlis gverdiT, 

Robis SigniT, idges wvriltaniani xe. dabali saxlis gverdiT, 

Robis SigniT, idges msxviltaniani xe. saxlebi SeaerTeT gziT. 

magram dabal saxlTan gza farTovdebodes, farTo iyos, maRal 

saxlTan _ viwrovdebodes, gza viwro iyos. Robis win daxateT 

soko didi qudiT da wvrili fexiT, dabali Robis win _ soko 

patara qudiT da msxvili fexiT. axla Robeebis gareT daxateT 

rame, rac Tqven gindaT. ras xatavT? 

24/4/!)ebyb{vmj*!lwbesbuj!tywbebtywb!nefcbsfpcbTj/! 

suraTi 13.3 (nakvTebi sxvadasxva ferebadaa gaferadebuli 

saSinao davalebaSi). _ am nakvTebSi yvelam erTad vipovoT

- 80 - 

kvadratebi da SemovxazoT! aba, romlebia kvadratebi? 

cxadia, Znelia im kvadratebis amocnoba, romlebic ceradaa 

daxazuli. amitom Tavidan, amocnobis gasaadvileblad, moswavleebma 

saWiro mimarTulebiT Semoabrunon rveulebi. masw.-i 

aCvenebs, rogor gaakeTon es. 

_ pirveli nakvTi aris Tu ara kvadrati? (Tan TiTiT 

miuTiTebs) {ki, namdvilad kvadratia} SemovxazoT! es meore 

nakvTi? aba SemovabrunoT rveuli da vnaxoT! {kvadrati 

yofila!} giglam mipasuxos: meore nakvTi Semobrunebul 

rveulze Turme kvadrati yofila. axla rveuli isev sworad 

davdoT. is meore nakvTi mainc kvadrati iqneba Tu ara? {ki, 

kvadrati kvadratad darCeba, rogorc ginda atriale naxazi!} 

meore nakvTic SemovxazoT! amaze ras ityviT? aba 

SemoatrialeT rveuli! {ara, araa kvadrati, imitom, rom 

maRalia!} axla danarCenebSi Tqven TviTon moZebneT kvadratebi 

da SemoxazeT! maS sul ramdeni kvadrati yofila? {xuTi}. 

24/5/ igive TamaSi, rac 12.4. 

g 14.!hbnfpsfcb/!sjhpcjUj!sjdywfcj/!nsbwbmlvUyfe/!hwfsefcj/! 

25/1/ masw.-i merxebs Soris Camoivlis da Seamowmebs: 13.5-s, 

13.6-is gaferadebis siswores; Tan gairCeva misi pasuxebi _ 

romelia meti da ratom. xolo 13.7 TavisTavad gairCeva 14.1-Si. 

25/2/!Ubwj!3/2.jt!Tfkbnfcb!eb!hbnfpsfcb/! 

jer ramdenime susti moswavle TvliT xmamaRla gaimeorebs 

ricxvTa rigs: erTi, ori, sami, oTxi, xuTi, eqvsi. 

amasobaSi masw.-i dafaze erT striqonSi erTmaneTisagan Sori- 

Sors daxatavs (anda, es winaswar eqneba momzadebuli) saTanado 

`dominosebr~ grovebs: 1, 2, 3, 4, 5, 6. moswavleebs areulad 

SeekiTxeba: _ es ramdenia? es ramdenia? ... romelia yvelaze 

bevri? yvelaze cota? esaa meti Tu es? es naklebia Tu es? 

Semdeg mTw-dan gameordeba oTxeulis, xuTeulisa da 

eqvseulis sxvadasxva dawyoba: ori da ori, sami da ori, xuTi 

da erTi da ase Semdeg. 

moswavleebi grovebs `kiTxuloben~ waRmac (marcxnidan

- 81 - 

marjvniv) da ukuRmac (marjvnidan marcxniv _ Tumca am 

sityvebis xseneba araa saWiro, unda iTqvas ùkuRma~ da TiTis 

moZraobiT mieTiTos). magaliTad, `5~-Si erTerT elifsSi 

`waikiTxaven~ _ sami da ori. Semdeg, ukuRma `wakiTxvisas~ _ 

ori da sami. 

_ ramdeni TiTi gvaqvs cal fexze? {xuTi} ramdeni TiTi 

aqvs qaTams? {oTxi} ramdeni fexi aqvs Jirafs? {oTxi} ramdeni 

fexi aqvs siraqlemas? {ori} ramdeni frTa aqvs SaSvs? {ori} 

ramdeni fexi aqvs futkars? {eqvsi}. 

Semdeg ganvixilavT saSinao davalebas 13.7 anu suraTs 14.1. 

_ ramdenia didi Tevzi? {xuTi} patara Tevzi? {eqvsi} ramdenia 

lurji Tevzi? {sami} yviTeli? {oTxi} wiTeli? {oTxi} 

ramdeni Tevzi micuravs qvemodan zemoT? {ori} zemodan 

qvemoT? {ori} uknidan winisken? {sami}. (es ukanaskneli 

SekiTxva cota Znelia, TavsatexiviTaa). 

25/3/! sjhpcjUj! sjdywfcj;! qjswfmj-! nfpsf-! nftbnf-! nfpUyf-! 

nfyvUf-!nffrwtf/!dafaze winaswar momzadebulia xuTsarTuliani 

saxlis naxati. 

masw.-i rig-rigobiT gamoiyvans eqvs moswavles da daayenebs 

erTmaneTis ukan. pirvelad gamosuli dadgeba win, mis ukan _ 

meore da ase Semdeg. 

_ ramdeni moswavle dgas aq? {eqvsi} vinaa pirveli? 

{TinaTini} meore? ... (da ase Semdeg). 

_ es saxli ramdensarTuliania? gamovides nukri da 

daTvalos! axla saTiTaod gviCvenos, romelia pirveli 

sarTuli; meore sarTuli... (da ase Semdeg). 

suraTi 14.2. masw.-i aCvenebs marcxena vardisfer wres. _ 

esaa pirveli wre. masSi dasviT erTi wertili. axla meore 

wreSi dasviT ori wertili, ise, rogorc mTw-Sia. axla meore 

wreSi dasviT sami wertili, ise, rogorc mTw-Sia. ... (da ase 

Semdeg) wavikiTxoT raodenobani: erTi, ori, sami, oTxi, xuTi, 

eqvsi. axla wavikiTxoT, wreebi meramdeneebia {pirveli, meore, 

mesame, meoTxe, mexuTe, meeqvse}. 

25/4/!obtxbwm!nsbwbmlvUyfeUb!hwfsefcj/!nbUj!ebUwmb/!

- 82 - 

masw.-i klass aCvenebs kubis erT waxnags. _ es ra aris? 

{kvadrati} es? (TiTs gaayolebs wibos) {monakveTia} am 

monakveTs hqvia kvadratis gverdi. gverdi. axla vnaxoT es 

dafaze daxazuli kvadrati. gamovides nodari da gviCvenos misi 

erTi gverdi. axla meore gverdic gviCvene. sul ramdeni gverdi 

aqvs kvadrats? {oTxi} kvadrats oTxi gverdi aqvs. 

suraTi 14.3. _ ai esaa pirveli nakvTi. ra aris es? 

{samkuTxedi} romelia meore nakvTi? {samkuTxedi} mesame? 

{kvadrati} meoTxe? {kvadrati} mexuTe? {xuTkuTxedi} meeqvse? 

{eqvskuTxedi} Savad gaxazeT pirveli samkuTxedis erTi gverdi. 

sul ramdeni gverdi aqvs samkuTxeds? {sami} axla Savad 

gaxazeT meore samkuTxedis samive gverdi. mesame ra nakvTia? 

{kvadrati} gaxazeT Savad misi erTi gverdi. meoTxe ra nakvTia? 

{isic kvadratia} gaxazeT Savad misi yvela gverdi. ramdeni 

gverdi aqvs kvadrats? {oTxi} mexuTe ra nakvTia? {xuTkuTxedi} 

gaxazeT Savad misi erTi gverdi. sul ramdeni gverdi 

aqvs xuTkuTxeds? {xuTi} meeqvse ra nakvTia? {eqvskuTxedi} 

gaxazeT Savad misi erTi gverdi. sul ramdeni gverdi aqvs eqvskuTxeds? 

{eqvsi} diax, samkuTxeds sami gverdi aqvs, kvadrats 

_ oTxi, xuTkuTxeds _ xuTi, eqvskuTxeds _ eqvsi. 

amis Semdeg moswavle daTvlis xolme rTuli formis mqone 

mravalkuTxedebis gverdebs roca dasWirdeba. 

25/5/!`nbubsfcmpcbob~/ gamoviyvanT erT moswavles. is ityvis: 

_ me var pirveli vagoni! _ da dadgeba SuaSi. gamoviyvanT sxva 

mos-wavles: _ Sen romeli vagoni unda iyo? {meore} diax, Sen 

iqnebi me-ore vagoni, aba dadeqi Sens adgilas! (dadgeba 

pirveli moswavlis ukan da welze mokidebs xelebs). erTi da 

erTi _ miviReT ori! gaimeoros naTiam! (naTia imeorebs, is ar 

gamogvyavs `samatareblod~!)... 

ase SevadgenT èqvsvagonian matarebels~. TviTeuli 

moswavlis damatebis Semdeg romelime mjdari moswavle ambobs: 

_ ori da erTi _ miviReT sami! (da ase Semdeg). 

_ matarebelo, iare! (bavSvebi daiZrebian da daiwyeben merxebs 

Soris siaruls). axla daviwyoT vagonebis moxsna! meeqvse

- 83 - 

vogono, moexseni matarebels! (moexsneba da dajdeba) eqvss 

moaklda erTi da dagvrCa... ramdeni? {xuTi} gaimeoros mariamma! 

(da ase Semdeg, sanam yvela moswavle dajdeba, Tan amboben: _ 

xuTs moaklda erTi da darCa oTxi! _ da ase Semdeg). 

g 15.!nprnfefcb!Tflsfcb/!{f/rwf!eb!sjh/!sjdyw/!yvUl/!eb!frwtl/! 

26/1/ masw.-i merxebs Soris Camoivlis da Seamowmebs: 14.5-s, 

14.6-is gaferadebis siswores; Tan gairCeva misi pasuxebi _ 

romeli nakvTi ramdenia. 14.7-s, cxadia, ar sWirdeba ganxilva. 

---------------------------------------------------------------------------------- 

J.4/! nprnfefcb! Tflsfcb/! mTavaria gaazreba imisa, Tu ra aris 

moqmedeba Sekreba. yvelaferi xdeba cifrul CanawerTa gareSe 

cifruli Canawerebi aq mxolod masw.-isTvisaa! bavSvisTvis gaazrebis 

erTaderTi gza ki aris sakuTari xeliT am ariTmetikuli 

moqmedebis Sesabamisi namdvili moqmedebis Catareba nivTebiT. 

ricxvTa Sekreba ariTmetikuli moqmedebaa, misi Sesabamisi 

nivTieri moqmedebaa grovaTa SeerTeba. swored es unda gakeTdes, 

Tanac mravaljer. 

am dros Sedegis daTvla anu jamis angariSi araviTar SemTxvevaSi 

ar unda moxdes! jer irkveva, ra aris moqmedeba Sekreba, da 

ara is, Tu rogor viangariSoT am moqmedebis Sedegi. 

Semdeg iswavleba zepiri Sekreba eqvsis farglebSi, nivTebiT 

Sekrebis niSnisa da cifrebis gareSe. ZiriTadi sayrdenia 

dominosebri TvalsaCinoeba. mizania eqvsis farglebSi ZiriTadi 

jamebis angariSi da pirveli damaxsovreba. 

SesakrebTa gadanacvlebadoba iswavleba yovelgvari terminebis 

gareSe mxolod konkretuli magaliTebis SemowmebiTa da 

amdagvari daskvniT: `sul erTia, rogor vityviT jami mainc igive 

iqneba~. yuradReba maxvildeba terminze ìgive~. 

sasurvelia, rom Sekrebis pirveli ramdenime gakveTilis 

ganmavlobaSi rac SeiZleba met moswavles da rac SeiZleba xSirad 

mouwioT nivTebis grovaTa sakuTari xeliT Sekreba. 

---------------------------------------------------------------------------------- 

26/2/!Tflsfcb-!wjUbsd!nprnfefcb!`!ojwUUb!hspwfcjU/!`Tflsj. 

cf~/!!masw.-s winaswar momzadebuli aqvs moZrav nivTTa grovebis

- 84 - 

wyvilebi. TviTeul grovaSi nivTebis raodenoba unda iyos 8-9 

an ufro meti. rogorc yovelTvis, aucilebelia moZravi 

geometriuli nakvTebis grovebi, gansakuTrebiT _ ukanasknelad 

naswavli xuTkuTxedebi da eqvskuTxedebi. garda amisa, kvadratebi 

aucileblad ceradac unda daewyos. 

masw.-i gamoiyvans sust moswavles da daavalebs: _ aba, es 

ori grova SeaerTe, Sekribe, SeaerTe! (moswavle xelebiT 

gaaerTianebs kvadratebsa da xuTkuTxedebs). diax, esaa Sekreba! 

gigam es grovebi Sekriba (meti gamokveTilobisaTvis masw.-i 

SeerTebul grovas Semoxazavs kidevac). axla gamovides naTela 

da es grovebic Sekribos (amis Semdeg sul vambobT `Sekreba~, 

`Sekribe~ da sxva, aRar vityviT `SeaerTe~). 

ase Seikribeba yvela momzadebuli wyvili grovaTa: kubebi, 

Cxirebi, rgolebi, eqvskuTxedebi, nuSebi, saTamaSo peplebi. 

Semdeg Sekreba moxdeba suraTze 15.1. _ eseni viTom 

Toliebia. aba aqeT, cariel adgilas, kvercxiviT xazi romaa 

(miuTiTebs elifsze) gadmoxateT yvela Tolia ise, rom es 

ori grova Seikribos! 

moswavleebi jer erTi grovis `Toliebs~ gadaxataven, 

Semdeg _ meore grovisas. masw.-i Seamowmebs, romelime xom ar 

gamorCeniaT, an SecdomiT orjer xom ar gadauxatavT. 

26/3/! {fnpU! /! rwfnpU! njnbsUfcjt! eblbwTjsfcb! sjhpcjU! sjd. 

ywfcUbo/! masw.-i dafa-magidaze erT svetad daawyobs sxvadasxva 

ferisa da zomis eqvs kubs, erTmaneTze. _ ramdeni kubia am 

grovaSi? {eqvsi} esaa kubebis eqvseuli. ai, qvemodan esaa 

pirveli kubi, didi lurji kubi. esaa qvemodan pirveli kubi. 

romelia qvemodan meore kubi? {didi yviTeli kubi} qvemodan 

mesame? {wiTeli kubi} ... (da ase Semdeg). 

_ es iyo qvemodan. axla vnaxoT zemodan! romelia zemodan 

pirveli kubi? {patara mwvane kubi} zemodan meore romelia? 

{Savi kubi} ... (da ase Semdeg). 

aseve ganixileba 15.2 suraTze daxazuli nakvTebis eqvseuli. 

SekiTxvebia: _ sul ramdeni nakvTia aq daxazuli? romelia 

yvelaze qveda nakvTi? yvelaze zeda? romelia qvemodan meore

- 85 - 

nakvTi? qvemodan meoTxe? meramdenea eqvskuTxedi? kvadrati? 

romeli nakvTia samkuTxedis qvemoT? xuTkuTxedis zemoT? 

romeli nakvTis qvemoTaa rgoli? romeli nakvTis zemoTaa 

kvadrati? (da ase Semdeg). 

rogorc yovelTvis, TviTeuli SekiTxva daesmis TiTo 

moswavles, romelic pasuxobs adgilidan, swrafad, fexze 

audgomlad, radgan yvelani msr-s akvirdebian. gunduri 

pasuxebi dauSvebelia! 

26/4/!yvUlvUyfej!eb!frwtlvUyfej!tywbebtywb!nefcbsfpcbTj/! 

jer gavimeorebT TvalsaCinoebis gareSe: _ ramdeni gverdi 

aqvs kvadrats? {oTxi} samkuTxeds? {sami} eqvskuTxeds? {eqvsi} 

xuTkuTxeds? {xuTi}. 

Semdeg gavimeorebT daxrili kvadratebis amocnobas: maswavlebeli 

dafaze daxazavs eqvs nakvTs, maT Soris mxolod zogi 

iqneba kvadrati (14.3 suraTis magvarad). moswavleebi 

amoicnoben kvadratebs da ityvian, igi meramdene nakvTia (Tvla 

mxolod marcxnidan marjvniv, masw.-i TiTiT aCvenebs mimar- 

Tulebas). daTvlian, sul ramdeni kvadrati yofila. 

suraTi 15.3. _ es pirveli nakvTi ra aris? {xuTkuTxedi} 

motrialebulia, magram mainc xuTkuTxedia. aba, movatrialoT 

rveuli da vnaxoT! meore nakvTi? {eqvskuTxedia}. axla moniSnuli 

wertilebis daxmarebiT Tqven daxazeT xuTkuTxedi da 

eqvskuTxedi! gverdiT rom carieli adgilia, iq Sin daxazavT! 

26/5/! Ùwbmjt! ebybnybnfcb~/ magnitur dafaze mikrulia 3 

kvadrati da 6 rgoli (`dominoseburi~ ganlagebiT, rogorc 

mTw-Sia), oRond dafas farda aqvs Camofarebuli. masw.-i erTi 

wamze naklebi droiT gadawevs fardas. moswavlem unda Tqvas, 

ra iyo dafaze {sami kvadrati da eqvsi rgoli}. dafaze 

icvleba rogorc nakvTebi, ise maTi raodenoba. meore wreze 

grovaTa ganlageba ar emTxveva `dominosebur~ ganlagebas, cota 

Secvlilia. mesameze _ ufro metadaa Secvlili da ase Semdeg. 

g 16.!bewjmj!Tflsfcb/!xjo/vlbo!eb!sjhpc/!sjdyw/!hbnfpsfcb/! 

27/2/!bewjmj!Tflsfcb;!1+1, 2+1, 1+2.

- 86 - 

jer gameordeba grovaTa Sekreba (grovebi winaswar unda 

iyos momzadebuli). sasurvelia, rom rac SeiZleba met moswavles 

mouwios nivTebis grovaTa sakuTari xeliT Sekreba. 

_ axla vnaxoT, ra ricxvs miviRebT, erTi da erTi rom 

SevkriboT (Seikribeba saTvleli nivTebiT). diax, erTi da erTi 

SevkribeT da miviReT ori. axla SevkriboT erTi da ori... ra 

ricxvi miviReT? ... ori da erTi... Semdeg gadaSlian mTw-s 

`1-2-3~-ze. _ moZebneT, sad SeiZleba davinaxoT erTisa da 

erTis Sekreba! orisa da erTis Sekreba? 

27/3/!xjo!/!vlbo!njnbsUfcjt!eblbwTjsfcb!sjhpcjU!sjdywfcUbo/!! 

ufsnjoj!`hbseb~/!masw.-i gamoiyvans eqvs moswavles da daayenebs 

erTmaneTis ukan-ukan. 

_ ramdeni moswavle dgas aq? {eqvsi} vinaa pirveli? 

{TinaTini} meore? ... (da ase Semdeg). Cven vnaxeT, vin iyo 

pirveli winidan. axla vnaxoT uknidan. vinaa pirveli uknidan? 

{nato} uknidan meore? (da ase Semdeg). 

Semdeg ganvixilavT suraTs 16.2. _ ramdeni kacunaa? {eqvsi} 

SeadareT erTmaneTs uknidan pirveli da uknidan meore kacuna. 

mipasuxos giorgim: riT gansxvavdebian isini erTmaneTisagan? 

(da ase Semdeg _ Sedardeba erTmaneTis momdevno kacunebis 

yvela wyvili). 

_ axla moZebneT winadan mesame kacuna. yvela kacunas, garda 

winadan mesamisa, qudi daaxureT! mesamis garda! (gamoikveTeba 

sityva `garda~). yvelas qudi unda exuros, oRond mesames _ 

ara! winidan mesamis garda! 

27/4/! Ubwj! 2/4.jt! Tfkbnfcb! eb! hbnfpsfcb/! suraTi 16.3 (unda 

iyos 6 ferad gaferadebuli). _ radara nakvTebi Cans am naxazze? 

{wertilebi, monakveTebi, wreebi, samkuTxedebi, kvadratebi, 

xuTkuTxedebi, eqvskuTxedebi} kidev? gamogrCaT erTi! {gareT 

rom didi rgolia!} ese igi, kidev rgolic yofila! rgolis 

gareT Tua rame? {ara} axla davTvaloT, ramden-ramdenia Tvi- 

Teuli nakvTi. ramdenia rgoli? {erTi} wertilebi? {eqvsi} monakveTebi? 

{xuTi} wreebi? {xuTi} wreebi imdenivea, ramdenicaa... 

ra? {ramdenicaa monakveTebi} diax, wreebisa da monakveTebis

- 87 - 

raodenoba tolia. samkuTxedebi ramdenia? {oTxi} kvadratebi? 

{xuTi} kvadratebi imdenivea, ramdenic... ra? {monakveTebi, wreebi} 

xuTkuTxedebi ramdenia? {erTi} xuTkuTxedi imdenivea, ramdenic... 

ra? {rgoli} eqvskuTxedebi ramdenia? {ori}. ra ufro 

meti yofila: eqvskuTxedebi Tu samkuTxedebi? {samkuTxedebi} 

kvadratebi Tu samkuTxedebi? {kvadratebi} (da ase Semdeg). 

_ axla rgolis gareT xeliT daxateT: eqvsi monakveTi; 

xuTi wertili; oTxi wre; sami samkuTxedi; ori kvadrati; 

erTi eqvskuTxedi. 

bolos, gavimeorebT TvalsaCinoebis gareSe: _ romel nakvTs 

aqvs oTxi gverdi? {kvadrats} sami gverdi? {samkuTxeds} eqvsi 

gverdi romel nakvTs aqvs? {eqvskuTxeds} xuTi gverdi? 

{xuTkuTxeds}. 

27/5/!`txsbgj!qbtvyfcj!`!tbqjsjtqjspoj~/ igive, rac 12.4. 

g 17.!njnbufcb/!hbnfpsfcb/!tywbebtywb!tbnlvUyfefcj/!ùpmj~. 

28/1/ kargad unda Semowmdes 16.6 da 16.7. 

28/2/!TfojTwob; unda davicvaT enobrivi sizuste. `mimateba~ esaa 

ariTmetikuli moqmedeba, `damateba~ ara. `damateba~ vixmaroT 

sayofacxovrebo viTarebaSi, magaliTad: damimate kidev fxali, 

wyali daumate, ori moTamaSe daemataT da sxva. xolo ariTmetikaSi 

vixmaroT `mimateba~ an `Sekreba~, gaaCnia. 

---------------------------------------------------------------------------------- 

njnbufcb-!wjUbsd!nprnfefcb/!njnbufcb!eb!Tflsfcb/!`njvnbuf~/! 

keTdeba igive, rac 15.1-Si, oRond terminebTan `Sekreba / 

Sekribe~ erTad Semodis termini `mimateba / miumate~. kvlav 

winaswar momzadebuli grovebi ikribeba. xan vambobT `Sekribe 

es da es~, xan _ àmas miumate es~. 

_ ra yofila mimateba? {mimateba igivea, rac Sekreba}. risi 

tolia ors mimatebuli erTi? {samisa} erTs mimatebuli erTi? 

{orisa} erTs mimatebuli ori? {samisa}. 

Semdeg Sesruldes mimateba suraTze 17.1 (iseve, rogorc 

iyo suraTze 15.1), cxadia, Sedegis daTvlis gareSe! 

28/3/! Ubwj! 3/3.jt! hbnfpsfcb! eb! Tfkbnfcb/! ganvixilavT mTw-s

- 88 - 

ydis suraTs. _ vin aris winidan pirveli? meore? (da ase 

Semdeg meeqvsemde). axla vnaxoT uknidan. vin aris sul 

boloSi, yvelaze ukan? {ku} uknidan meore? mesame? (da ase 

Semdeg meeqvsemde). 

Semdeg masw.-i dasvams SekiTxvebs naswavl sivrciT 

mimarTebaTa Sesaxeb: _ weros ufro msxvili fexebi aqvs Tu 

cxens? nukrsa Tu xbos? baWias ufro farTo yurebi aqvs Tu 

cikans? pepelas frTa ufro sqelia Tu mamlis bibilo? cugas 

ufro wvrili fexebi aqvs Tu bokvers? melas kudi ufro 

sqelia Tu ciyvisa? mela ufro dabalia Tu beli? fiso ufro 

winaa Tu dedali? ixunia ufro maRlaa Tu Citi? (da ase 

Semdeg _ saWiroa mravali SekiTxva). sul ramdeni pepelaa? 

{xuTi}. sul ramdeni frinvelia? {eqvsi}. romlebia frinvelebi? 

{dedali, mamali, Citi, bu, ixvis WuWuli, wero}. tolia Tu 

ara peplebisa da frinvelebis raodenoba? {ara}. romlis 

raodenobaa meti? {frinvelebia 6, peplebi _ 5, frinvelebi 

metia erTiT}. 

28/4/! tywbebtywbhwbsj! tbnlvUyfefcj/! ufsnjofcj! `fsUobjsj~-! 

ùpmj~-! `nypmpe~/! dafaze daxazulia sul sxvadasxvagvari 

samkuTxedebi, xuTi cali da agreTve sxva nakvTebi (17.7-is magvarad, 

oRond erT striqonSi). erToblivi msjelobiT saTiTaod 

gairkveva, romeli maTgania samkuTxedi da romeli _ ara. 

masw.-i moiTxovs dasabuTebas: _ saidan ici, rom es 

samkuTxedia? {sami gverdi aqvs} es ratom ar aris samkuTxedi? 

(da iseTive nakvTs daxazavs dafaze) {imitom, rom oTxi gverdi 

aqvs} es ratom ar aris samkuTxedi? (da iseTive nakvTs 

daxazavs dafaze) {imitom, rom aq momrgvalebuli aqvs, 

samkuTxeds ki momrgvaleba arsad ara aqvs} (da ase Semdeg). 

_ maS, ramdeni samkuTxedi yofila dafaze daxazuli? 

{xuTi} es xuTive _ samkuTxedebia. magram aris Tu ara isini 

erTnairebi? {ara, yvela sxvadasxvanairebia!} axla davxazoT 

iseTive samkuTxedebi (masw.-i daxazavs ramdenimes). aba romelia 

am samkuTxedis toli samkuTxedi? gamovides aleko da 

gvaCvenos! diax, es da es samkuTxedebi erTnairebia, tolebia.

- 89 - 

es samkuTxedi ki romlis tolia? (da ase Semdeg). 

Semdeg 17.3-Si daxazaven or sxvadasxvagvar samkuTxeds. 

danarCenebs da cariel adgilas samkuTxedebs Sin daxazaven. 

termini `mxolod~ Semogvaqvs 17.7-Si. 

28/5/! `txsbgj! qbtvyfcj~/ SekiTxvebi igivea, rac 17.2-Si, 

magram warmodgeniT, TvalsaCinoebis gareSe. 

28/7/! tjbymfb! ufsnjoj! `nypmpe~-! mas masw.-i sagangebod 

ganmartavs: _ unda gaaferadoT mxolod samkuTxedebi, marto 

samkuTxedebi, sxva araferi ar unda gaaferadoT! 

siaxlea agreTve logikuri kavSiri àn~, magram mas aq ar 

sWirdeba ganmarteba _ Sin vinc rogor gaigebs, ise Seasrulos. 

g 18.!Tflsfcb/!tbzpg/!bnpd/!sbpe/!npnbufcb{f/!tbnlvUy/!yb{wb/! 

29/1/ 17.7-Si kvlav gamoikveTeba sityva `mxolod~. garda 

amisa, gairkveva, vin rogor gaigo `wiTlad an mwvaned~ {ginda 

wiTlad, ginda mwvaned, ase SeiZleboda da isec} mTavari sxva 

ramea: àn~ niSnavs, rom moswavles SeeZlo zogi samkuTxedi 

wiTlad gaeferadebina, zogi ki _ mwvaned, anda _ yvela erT 

ferad. am mxrivac gasarkvevia. 

TfojTwob; davalebaSi rom yofiliyo `wiTlad da mwvaned~ _ 

maSin saWiro iqneboda TviTeuli samkuTxedis erTdroulad 

orive ferad gaferadeba. 

17.8-Si Semowmdes isic, nakvTebi Tu aqvT gaferadebuli. 

29/2/! Tflsfcb: 3+1, 1+3, 2+2. TviTeuli es Sekreba jer Sesruldeba 

sxvadasxva saTvleli nivTebiT (umjobesia, magnitur 

dafaze). moswavleebi daTvlian, rom Sekrebis (mimatebis) 

Sedegad viRebT oTxs. 

Semdeg _ mTw `4~. _ ramdeni wrea aq? {oTxi} ramdeni 

wrea aq? {oTxi} rogoraa wreebi dalagebuli? {sami da erTi} 

samisa da erTis SekrebiT ramdeni miiReba? {oTxi} aq es Tu 

dainaxeT? rogor dainaxeT? {oTxi wre dalagebulia ase: sami 

da erTi} axla davTvaloT wreebi marjvnidanac (Tan TiTis 

moZraobiT uCvenebs mimarTulebas). rogoraa dalagebuli? 

{erTi da sami} erTisa da samis SekrebiT ramdeni miiReba?

- 90 - 

{oTxi} aq es Tu dainaxeT? ... _ oTxi wre rogoraa dawyobili 

aq? {ori da ori} orisa da oris SekrebiT ramdeni miiReba? 

{oTxi} aq es Tu dainaxeT? ... 

sxvadasxva moswavleebs, ramdenjerme, sruli pasuxis moTxovniT: 

_ ramdenia sams mivumatoT erTi? erTs mivumatoT 

ori? erTs mivumatoT sami? erTs mivumatoT erTi? ors 

mivumatoT ori? ors mivumatoT erTi? 

--------------------------------------------------------------------------------- 

JJ.4/! jxzfcb! tbzpgbdypwsfcp! bnpdbofcjt! Ufnb/ esaa umartives 

sayofacxovrebo viTarebaTa maTematikuri modelireba: Sekrebagamoklebis 

moqmedebebiT. ese igi, saWiroa sayofacxovrebo 

viTarebis Sesabamisi ariTmetikuli modelis ageba. es araa Zneli 

imdenad, ramdenadac masw.-ma es viTareba sqematurad unda gaaTvalsaCinoos. 

magaliTad, amocanaSi sityvierad mocemulia totze 

mjdari 3 Citi. masw.-i gaaTvalsaCinoebs magidaze dadebuli 3 

kenWiT. Semdeg mofrinda kidev 2 Citi. masw.-i daumatebs kidev 2 

kenWs. ufro meti gaTvalsaCinoba (magaliTad, saTamaSo Citunebis an 

Citebis suraTebis dawyoba) araTu zedmetia, aramed uaresicaa 

moswavlis azrovneba Citis xatebas miejaWveba, rac daabrkolebs 

maTematikur ganyenebas. [ix. zms $ 6] 

TandaTan daviwyebT amgvari amocanebis amoxsnas yovelgvari 

gaTvalsaCinoebis gareSec. sqematur gaTvalsaCinoebas gamoviyenebT 

mxolod im SemTxvevaSi, Tuki romelime moswavles gauWirdeba 

viTarebis gonebaSi warmodgena (Tumca, mogvianebiT, roca daiwyeba 

ormoqmedebiani amocanebi, sqemis daxatva aucilebeli iqneba). sqema 

dafaze ixazeba. moswavleebisaTvis gasagebad amas vuwodoT 

àmocanis daxatva, viTom es bebiaa, es papa~. 

`viTom~ bavSvisaTvis bunebrivia da advilad misaRebi. 

es yovelive araa Zneli dasauflebeli. SedarebiT ufro Znelia 

is, rom klasma kargad gaiazros, Tu ra aris `-iT meti~ da `-iT 

naklebi~ sityvierad, da ara mxolod TvalsaCino doneze. amas 

ramdenime sagakveTilo monakveTs davuTmobT (21.2-dan). 

Tavidan Zalian advili amocanebia. amitom mTavaria ara pasuxis 

angariSi, aramed amocanis amoxsnis unarCvevis Camoyalibeba. amis-

- 91 - 

Tvis saWiroa oTxi safexuri, romelime maTganis gamoutoveblad! 

1) amocanis pirobis kargad gaazreba; 

2) ariTmetikuli modelis Seqmna anu saWiro moqmedebaTa 

dagegmva (SesaZloa, sqemasTan erTad); 

3) moqmedebaTa Sesruleba anu pasuxis angariSi; 

4) pasuxis Camoyalibeba. 

Zalian uxeSi Secdomaa, roca amocanis amoxsna dayvanilia 

mxolod mesame safexurze. pirveli da meore safexurebi ufro 

mniSvnelovania, vidre mesame! 

masw.-s unda axsovdes: ar gvainteresebs amocanis arc pasuxi da 

arc moqmedebani manam, sanam misi piroba ar gaazrebula. amitom 

amocanis amoxsnas yovelTvis misi pirobis gameoreba uZRvis, Semdeg 

ki sqemis Sedgena. 

--------------------------------------------------------------------------------- 

29/3/!vnbsujwftj!tbzpgbdypwsfcp!bnpdbofcj!sbpefopcjt!np. 

nbufcb{f/!!_ axla me gvetyviT erT sul advil amocanas. SoTam 

yanidan sami gogra moitana (dafaze daixazeba sami rgoli). 

Semdeg gavida da kidev erTi gogra moitana. (gverdiT miexateba 

erTi rgoli). sul ramdeni gogra moutania SoTas? ara, nu 

CqarobT! pasuxi jer ar minda! jer gavimeoroT amocanis piroba 

(am sityvis gamokveTiT). vin gaimeorebs? 

gameorebisas, roca moswavle ityvis `sami gogra~, masw.-i 

TiTs daadebs dafaze sam rgols da CarCoTi Semoxazavs maT. 

roca moswavle ityvis `kidev erTi gogra~, masw.-i TiTs daadebs 

dafaze erT rgols. 

Semdeg moswavleebi albaT pirdapir pasuxs ityvian _ oTxi. 

maSin masw.-i hkiTxavs: _ saidan ici? {sams mivumate erTi da 

miviRe oTxi}. maS ra moqmedeba unda SevasruloT? {mimateba}. 

axla Savlegi gvetyvis pasuxs, oRond sruli pasuxiT! {SoTas 

sul oTxi gogra moutania}. 

aseve safexurebad gairCeva amocana: saxlis win, kordze, 

ori fiWvi idga (masw.-i dafaze or mokle xazs Camousvams). 

gazafxulze kidev oTxi naZvi darges (oTxi xazi dafaze). 

ramdeni xe gaxda kordze?

- 92 - 

_ ramdeni amocana amovxseniT Cven ukve? {ori} axla kidev 

erT amocanas getyviT. meramdene amocana iqneba es? {mesame} 

diax, axla mesame amocana. naTias sami didi saTamaSo hqonda, 

naxeT am suraTze (18.2). SemoxazeT es sami didi saTamaSo: 

Tojina da ori daTunia! dabadebis dReze kidev aCuqes didi 

Tojinebi. naxeT suraTze: ramdeni Tojina uCuqebiaT 

naTiasTvis? {ori} diax, oric aCuqes. naTias Tojinebi moemata 

Tu moaklda? {moemata} ramdeni Tojina gauxda naTias? jer 

unda gavimeoroT amocanis piroba... (iseve amoixsneba). 

29/4/!tywbebtywbhwbsj!tbnlvUyfefcjt!ebyb{wb/! 

masw.-i dafaze aCvenebs, Tu rogor SeiZleba sxvada-sxvagvar 

samkuTxedTa daxazva: winaswar wertilebis sameulis moniSvna 

(gamoikveTeba gamoTqma `wertilebis sameuli~) da Semdeg maTi 

SeerTeba saxazavis saSualebiT. 

18.3-is patara (marcxena) CarCoSi moswavleebi Caxazaven (da, 

cxadia, gaaferadeben) or samkuTxeds, isini sxvadasxvagvari 

unda iyos. _ gaaferadeT mwvaned an (am sityvis gamokveTiT) 

yviTlad! (vnaxoT, rogor gaigeben àn~ kavSirs). 

_ ra SeiZleba davxatoT kvadratis gamoyenebiT? {saxli} 

samkuTxedis gamoyenebiT? {droSa, naZvi, saxlis saxuravi} wris 

gamoyenebiT? {mze, borbali} hoda cariel did CarCoSi CaxateT 

xeliT: patara kvadrati _ es viTom ra aris? {saxli} zed daaxureT 

samkuTxedi, viTom ra aris? {saxlis saxuravi} saxuravze 

daxazeT patara monakveTi, viTom joxia da zed daamagreT 

sul patara samkuTxedi, viTom ra aris? {droSa} saxlis 

gverdiT daxateT viwro maRali samkuTxedi (aCvenebs dafaze), 

viTom ra aris? {naZvi} naZvs qvemoT tanic sWirdeba, ra mivuxazoT? 

{monakveTi} naZvis qvemoT daxateT wre, viTom ra aris? 

{borbali} saxlis zemoTac daxateT wre, viTom ra aris? {mze} 

mzes sxivebi sWirdeba, ra davuxatoT? {monakveTebi}. 

29/5/!`txsbgj!qbtvyfcj!`!lblvo.ubTfcj~/ masw.-i daakakunebs an 

taSs Semokravs ramdenimejer (araumetes eqvsjer) da moswavlem 

unda Tqvas, ramdeni iyo. meore wridan Zneldeba: masw.-i kidec 

akakunebs da taSsac ukravs ise, rom maTi raodenobaTa ja-

- 93 - 

mi ar aRematebodes eqvss. magaliTad, sami taSi da ori dakakuneba 

{xuTi}, anda erTi dakakuneba da xuTi taSi {eqvsi}. 

g 19.!Tflsfcb/!tbz/!bnpd/!Tflsfcb{f/!tywbebtywb!pUylvUy/!yb{wb/! 

2:/1/ amocana 18.7 _ SemoqmedebiTia. masw.-s uyuradRebod ar 

unda darCes arcerTi moswavliseuli amoxsna. yvela 

saintereso da lamazi naWrela dafaze unda daxazos, xolo 

moswavle Seaqos. saWiroa wamaxalisebeli niSnis dawerac. 

TfojTwob; SemdgomSi amgvar SemTxvevebSi mivaniSnebT mxolod 

mokled: amocana SemoqmedebiTia! masw.-s ecodineba, ra unda 

gaakeTos amgvar SemTxvevebSi. 

2:/2/!Tflsfcb: 4+1, 1+4, 3+2, 2+3.!yvelaferi iseve, rogorc 18.1. 

TviTeuli es Sekreba jer Sesruldeba sxvadasxva saTvleli 

nivTebiT (umjobesia, magnitur dafaze). moswavleebi daTvlian, 

rom Sekrebis (mimatebis) Sedegad viRebT xuTs. 

Semdeg _ mTw `5~. _ ramdeni wrea aq? {xuTi} ramdeni 

wrea aq? {xuTi} rogoraa wreebi dalagebuli? {sami da ori} 

samisa da oris SekrebiT ramdeni miiReba? {xuTi} aq es Tu 

dainaxeT? rogor dainaxeT? {xuTi wre dalagebulia ase: sami 

da ori} axla davTvaloT wreebi marjvnidanac (Tan TiTis 

moZraobiT uCvenebs mimarTulebas). rogoraa dalagebuli? {ori 

da sami} orisa da samis SekrebiT ramdeni miiReba? {xuTi} aq 

es Tu dainaxeT? ... 

_ oTxi wre rogoraa dawyobili aq? {erTi da oTxi} 

erTisa da oTxis SekrebiT ramdeni miiReba? {xuTi} aq es Tu 

dainaxeT? ... 

sxvadasxva moswavleebs, ramdenjerme, sruli pasuxis 

moTxovniT: _ ramdenia sams mivumatoT ori? erTi? erTs 

mivumatoT oTxi? ori? erTs mivumatoT sami? erTs mivumatoT 

erTi? ors mivumatoT ori? ors mivumatoT sami? ors 

mivumatoT erTi? sams mivumatoT ori? 

2:/3/!vnbsujwftj!tbzpgbdypwsfcp!bnpdbofcj!Tflsfcb{f!`!kb. 

njt! np[fcob/! am safexurze amocanaTa piroba srulad 

gaTvalsaCinovdeba. _ mousmineT amocanas! zurabma rveulSi

- 94 - 

daxaza ori kvadrati da oTxi wre (dafaze daixazeba). sul 

ramdeni nakvTi dauxazavs zurabs? jer ra unda vqnaT? 

{gavimeoroT piroba}... 

safexurebad amoixsneba agreTve kidev sami amocana, romelTa 

naxatebi mocemulia suraTze 19.2. _ erT totze oTxi Citi ijda, 

meoreze _ erTi. ramdeni Citi mjdara orive totze erTad? 

amocana Cveulebrivad amoixsneba, oRond bolos daemateba 

erTi ram: mis pasuxs (xuTi) moswavleebi `dominoseburad~ 

daxataven `dominoseburi~ oTxeulisa da erTeulis gverdiT. 

_ erT qvaze sami lokokina ijda, meorezec _ amdenive (am sityvis 

gamokveTiT). ramdeni lokokina mjdara orive qvaze erTad? 

amoxsnisas saWiroa damatebiTi SekiTxva: ramdeni lokokina 

mjdara meore qvaze? saboloo pasuxi daixateba `dominoseburad~. 

_ erT Taroze ido... ramdeni wigni? daTvaleT! {oTxi} 

meoreze... ? {ori} ramdeni wigni yofila orive Taroze erTad? 

saboloo pasuxi daixateba `dominoseburad~. 

_ sul ramdeni amocana amovxseniT dRes? {oTxi} raze iyo 

pirveli amocana? {nakvTebis daxatvaze} meore amocana? 

{Citebze} mesame? {lokokinebze} meoTxe? {wignebze} ra 

moqmedeba SevasruleT am amocanebis amosaxsnelad? {Sekreba}. 

diax, oTxive amocanis amosaxsnelad saWiro yofila Sekreba! 

axla vinme xom ver moigonebs axal amocanas, romlis 

amosaxsnelad Sekreba iqneba saWiro? 

_ vinc SeZlebs, Sin moigonos kidev aseTi axali amocana. 

xval mogviyves. 

es ar iqneba savaldebulo davaleba. visac ar daaviwydeba 

da SeZlebs, momdevno gakveTilze masw.-i mas Seaqebs da 

wamaxalisebel niSans dauwers. 

2:/4/! tywbebtywbhwbsj! pUylvUyfefcj; bnp{ofrjmj! bo! Db{of. 

rjmj/!nbUj!ebyb{wb/!!!!jer gavimeorebT TvalsaCinoebis gareSe: 

_ ramdeni gverdi aqvs kvadrats? {oTxi} samkuTxeds? {sami} 

eqvskuTxeds? {eqvsi} xuTkuTxeds? {xuTi}. 

Semdeg masw.-i dafaze daxazavs erT arasimetriul 

oTxkuTxeds (daaxloebiT imdagvars, rogoricaa 19.3-is pir-

- 95 - 

veli oTxkuTxedi). gverdiT miuxazavs erT arasimetriul 

xuTkuTxeds. _ vin mixvdeba, es nakvTi ra nakvTia? 

{oTxkuTxedi} saidan mixvdi? {oTxi gverdi aqvs} marTalia, 

oTxkuTxeds oTxi gverdi aqvs. es Tu aris oTxkuTxedi? {ara, 

ar aris} ratom? {xuTi gverdi aqvs} axla daakvirdiT am 

suraTs (19.3). wiTlebi ra nakvTebia? {oTxkuTxedebi}. axla 

daxazeT kidev ori oTxkuTxedi. gaaferadeT! ra nakvTebia? 

{oTxkuTxedebi}. yvela? {diax, yvela} es yvela _ oTxkuTxedia. 

ratom? {yvelas oTxi gverdi aqvs}. 

Sin wiTeli nakvTebis qvemoT daxazeT iseTive oTxkuTxedebi. 

iseTive! (am sityvis gamokveTiT). boloSi carieli 

adgilia. iq daxazeT axali, sxvagvari oTxkuTxedebi, Sin Tqven 

TviTon moigoneT sul sxvagvarebi! 

SeniSvna.! maTematikaSi Cazneqil mravalkuTxeds ewodeba 

araamozneqili. es terminebi mxolod masw.-isTvisaa. moswavles 

ar moeTxoveba, rom erTmaneTisagan ganarCios amozneqili da 

araamozneqili mravalkuTxedebi! 

2:/5/!Tfkjcsj!tjnbSmjt!njyfewjU!ebehpnbTj _ ix. 5.4. 

2:/7/ saWiroa kidev gameoreba: Tu ra moeTxovebaT moswavleebs. 

ar unda gamorCeT dajdomis arcerTi SesaZlebloba. 

g 20.!Tflsfcb/!UwbmtbD/!`.jU!nfupcb~.!yvUlvUy/!frwtlvU/!yb{wb/! 

31/1/ 2:/7-Si erTi sirTulea: daxazul oTxkuTxedebs Soris 

erTi kvadraticaa. masw.-i amaze miaqcevinebs yuradRebas: _ es 

yvela _ oTxkuTxedia, magram maTgan erTi _ kvadratia. 

moZebneT igi da SemoxazeT! diax, es kvadratia. aris Tu ara 

kvadrati oTxkuTxedi? {ki} ratom? {imitom, rom kvadrats 

oTxi gverdi aqvs} diax, raki oTxi gverdi aqvs, amitom 

oTxkuTxedia! 

2:/8-is garCevisTvis misi magvari sqematuri naxati winaswar 

unda iyos momzadebuli dafaze. 19.8 _ SemoqmedebiTia! 

31/2/! Tflsfcb: 5+1, 4+2, 3+3, 2+4, 1+5.! TviTeuli es Sekreba 

jer Sesruldeba sxvadasxva saTvleli nivTebiT (umjobesia,

- 96 - 

magnitur dafaze). moswavleebi daTvlian, rom Sekrebis 

(mimatebis) Sedegad viRebT eqvss. 

Semdeg _ mTw `6~. gagrZeleba _ yvelaferi iseve, rogorc 

18.1 da 19.1. Sedegebs aucileblad sityvierad CamovayalibebinebT. 

31/3/! UwbmtbDjope! npdfnvm! sbpefopcbUb! `.jU! nfupcb~!`! 

nbtx/.jt!njfs!DwfofcjU!eb!nptxbwmjt!njfs!ebxzpcjU/!!gavimeorebT 

sityvas ìmdenive~. dafamagidaze awyvia 6 girCa. _ sandro! 

modi, Sen daawyve amdenive kubi! daTom ki dafaze daxatos 

amdenive samkuTxedi. kubebis raodenoba da girCebis raodenoba.... 

ra? {tolia}. gaimeoros salomem! kubebi imdenivea, ramdenicaa... 

{girCebi}. gaimeoros marinem. samkuTxedebi imdenivea, 

rac... ra? {girCebi, kubebi} samkuTxedebis raodenoba tolia... 

Semdeg masw.-i dafamagidaze dadebs kubebs. _ ramdeni kubia 

magidaze? {magidaze xuTi kubia} vis SeuZlia magidaze dados 

xuTze oriT meti raodenobis kubebi? 

masw.-i aclis bavSvebs, mixvdnen, xuTze oriT meti raodenobis 

kubebi rogor dadon, mousmens maT azrs. Semdeg axsnis: _ 

imisaTvis, raTa xuTze oriT meti raodenobis kubebi davdoT, 

jer unda davdoT xuTi kubi, imdenive kubi, ramdenic magidazea, 

da mere kidev _ ori kubi. 

rig-rigobiT gamoiyvans moswavleebs da daawyobinebs raimeze 

oriT meti raodenobis Cxirebs, kvadratebs, wreebsa da sxva. 

aseve moviqceviT samiT, oTxiT, xuTiT meti nivTebis dadebisas. 

cxadia, Sedegis daTvla araa saWiro. 

_ vinc SeZlebs, Sin moigonos axali amocana, romlis 

amosaxsnelad saWiro iqneba Sekreba. vinc moigonebs, xval 

mogviyves. 




31/4/! tywbebtywbhwbsj! yvUlvUyfefcj! eb! frwtlvUyfefcj! 

bnp{ofrjmj!bo!Db{ofrjmj!`!wbstlwmbwjtfcvsj/!nbUj!ebyb{wb/! 

masw.-i dafaze erT striqonSi daxazavs eqvs nakvTs: arasimetriul 

oTxkuTxeds, kvadrats, arasimetriul xuTkuTxeds, ara-

- 97 - 

simetriul eqvskuTxeds, rombs da erT Cazneqil xuTkuTxeds. 

_ ramdeni nakvTia dafaze? {eqvsi} pirveli nakvTi ra 

nakvTia? {oTxkuTxedi} ratom? {oTxi gverdi aqvs} meore? 

{kvadratia, Tan oTxkuTxedicaa} diax, Cven xom viciT, rom 

kvadrati Tan oTxkuTxediacaa! mesame ra nakvTia? {xuTkuTxedi} 

saidan iciT? {xuTi gverdi aqvs} diax, raki xuTi gverdi aqvs, 

amitom xuTkuTxedia. meoTxe ra nakvTia? {eqvskuTxedi} saidan 

iciT? {eqvsi gverdi aqvs} diax, raki eqvsi gverdi aqvs, amitom 

eqvskuTxedia. meeqvse ra nakvTia? {xuTkuTxedi} saidan iciT? 

{xuTi gverdi aqvs} diax, raki xuTi gverdi aqvs, amitom 

xuTkuTxedia. ramdeni oTxkuTxedi yofila dafaze? {sami} 

xuTkuTxedi? {ori} eqvskuTxedi? {erTi}. 

_ axla daakvirdiT suraTs (20.3). ramdeni wiTeli nakvTia? 

{oTxi} pirveli ra nakvTia? {xuTkuTxedi} ratom? {xuTi 

gverdi aqvs} diax, raki xuTi gverdi aqvs, amitom xuTkuTxedia. 

meore ra nakvTia? {eqvskuTxedi} ratom? {eqvsi gverdi aqvs} 

diax, raki eqvsi gverdi aqvs, amitom eqvskuTxedia (da ase 

Semdeg danarCen orzec). 

_ axla Tqven daxazeT da gaaferadeT erTi xuTkuTxedi da 

erTi eqvskuTxedi. oRond xuTkuTxedi gaaferdeT wiTlad, 

eqvskuTxedi ki _ mwvaned. jer daxazeT. mere gaferadeba ar 

SegeSaloT! (danarCenebs _ Sin daxazaven, cariel adgilas _ 

erT axalsac). 

31/5/! `txsbgj! qbtvyfcj~! `! ÙjUfcjt! Dwfofcb~/ masw.-i 

moswavles usaxelebs ricxvs (2, 3, 4, 5, an 6). moswavlem unda 

aCvenos amdeni TiTi, oRond aucileblad orive xelis TiTebi 

_ ise, rom erTad iyos dasaxelebuli ricxvi. magaliTad, Tuki 

masw.-ma daasaxela 4, moswavlem unda uCvenos 2 da 2 TiTi, an 1 

da 3 TiTi. 

Tavidan masw.-i daexmareba ase: _ gaixseneT mTw da rogorc 

iqaa, ise SekribeT ricxvebi. magaliTad: me giTxariT 5. 

rogoraa aq dalagebuli xuTi wre? {sami da ori} hoda, Tqvenc 

SegiZliaT miCvenoT sami TiTi da ori TiTi, ai, ase (aCvenebs). 

g 21.!Tftbls/!hbebobdwm/ `.jU!nfupcb~!`!ebxzpcb/!ebkhvgfcb/!

- 98 - 

32/2/!TftblsfcUb!hbebobdwmfcbepcb!UwjU!bn!ufsnjoUb!hbsfTf;!! 

5+1 = 1+5, 4+2 = 2+4, 1+3 = 3+1, 2+3 = 3+2, 1+2 = 2+1 `!zwfmb! 

obtxbwmj!TfnUywfwb/!suraTi 21.1. _ gogonas aqvs xili. ramdeni 

vaSli da ramdeni msxali aqvs gogonas? {oTxi vaSli da ori 

msxali} mis Zmas ra xili aqvs? {ori msxali da oTxi vaSli} 

sul ramdeni cali xili aqvs gogonas? {eqvsi} mis Zmas? {masac 

eqvsi} Zmas imdenive xili aqvs, ramdenic _ das. oTxi da ori 

xom imdenivea, rac ori da oTxi! 

magnitur dafaze momzadebulia `dominoseburi~ sameuli da 

gverdiT wyvili. 

gamoZaxebuli moswavle krebs, masw.-i ityvis: _ sams 

mivumatoT ori tolia xuTisa. modiT, axla davdoT da ise 

SevkriboT (oreuls dadebs sameulis win). ors mivumatoT sami, 

ramdenia? (gamoZaxebuli moswavle krebs, Tvlis da ambobs: _ 

ors mivumatoT sami tolia xuTisa). 

masw.-is daxmarebiT mTw gadaiSleba `6~-ze. 

_ ramdeni wrea am eqvskuTxedSi? {eqvsi} ramdeni wrea aq? 

(masw.-i uTiTebs) {eqvsi} rogoraa dawyobili wreebi? {xuTi da 

erTi} ramdeni miiReba xuTis da erTis SekrebiT? {eqvsi} diax, 

xuTs mimatebuli erTi udris eqvss. 

aq pirvelad vaxsenebT sityvas ùdris~. masw.-i xandaxan, roca 

dasWirdeba xolme, axsenebs am sityvas, moswavleebma SeiZleba 

sul ar ixmaron da Tqvan `tolia~ _ rogorc exerxebaT. 

_ rogor SevamowmoT, rom xuTs mimatebuli erTi namdvilad 

eqvsia? {davTvaloT es wreebi: xuTi da erTi _ sul eqvsi} 

axla miTxariT, erTisa da xuTis SekrebiT ramdeni miiReba? 

{eqvsi} saidan iciT? {ukuRma davTvaloT es wreebi: erTi da 

xuTi _ sul eqvsi} diax, maS erTs mivumatoT xuTi igivea, rac 

xuTs mivumatoT erTi. sul erTia, rogor SevkrebT: Tu gvinda, 

xuTs mivumatoT erTi, Tu gvinda, erTs mivumatoT xuTi. 

_ aq rogoraa dawyobili eqvsi wre? {oTxi da ori} (da 

yvelaferi iseve gameordeba, rogorc 5+1-isTvis)... diax, maS 

oTxs mivumatoT ori igivea, rac ors mivumatoT oTxi. sul 

erTia, rogor SevkrebT: gvinda, oTxs mivumatoT ori, gvinda,

- 99 - 

ors mivumatoT oTxi. 

_ axla miTxariT, risi tolia sams mivumatoT ori? 

{xuTisa} axla ki Zalian Cqara mipasuxeT, risi tolia ors 

mivumatoT sami! {esec xuTia!} diax, ors mivumatoT sami igivea, 

rac sams mivumatoT ori. sul erTia, rogor SevkrebT: gvinda, 

ors mivumatoT sami, gvinda, sams mivumatoT ori. 

_ axla miTxariT, risi tolia sams mivumatoT erTi? 

{oTxis} axla ki Zalian Cqara mipasuxeT, risi tolia erTs 

mivumatoT sami! {esec oTxia!} diax, erTs mivumatoT sami 

igivea, rac sams mivumatoT erTi. sul erTia, rogor SevkrebT: 

gvinda, erTs mivumatoT sami, gvinda, sams mivumatoT erTi. 

_ axla miTxariT, risi tolia oTxs mivumatoT erTi? 

{xuTisa} axla ki Zalian Cqara mipasuxeT, risi tolia erTs 

mivumatoT oTxi! {esec xuTia!} diax, erTs mivumatoT oTxi 

igivea, rac oTxs mivumatoT erTi. sul erTia, rogor 

SevkrebT: gvinda, erTs mivumatoT oTxi, gvinda, oTxs 

mivumatoT erTi. 

aseve ganixileba 1+2 = 2+1. 

32/3/!sbpefopcbUb!`.jU!nfupcb~!`!nptxbwmjt!njfs!ebxzpcb/! 

swrafad keTdeba ramdenime savarjiSo `-iT metobaze~ (ix. 

20.2). Semdeg erTi siaxle Semodis: masw.-i dafaze daxatavs 

raime ricxvis (erTidan Svidamde, magaliTad, xuTis) `dominos~ 

da moswavleebs daavalebs, dafazeve daxaton xuTze oriT meti 

wre. moswavle gadaxatavs `dominosebur~ xuTeuls da 

daumatebs kidev or wres. aseve _ samiT, oTxiT metobaze. 

Semdeg kidev erTi winsvla. masw.-i aRarafers xatavs, 

mxolod zepirad ambobs: _ daawyve Svidze oTxiT meti 

raodenobis kubebi. dafaze daxate xuTze xuTiT meti 

raodenobis oTxkuTxedebi. daxate Svidze oTxiT meti 

raodenobis monakveTebi! (da ase Semdeg). 

_ cariel CarCoSi (21.2) CaxateT xuTze oriT meti raodenobis 

patara samkuTxedi! sanam xatvas daiwyebdeT, kidev gavimeoroT: 

ra unda vqnaT? {unda davxatoT jer xuTi samkuTxedi 

da Semdeg davumatoT kidev ori samkuTxedi} axla daxateT!

- 100 - 

Sedegis daTvla (sul ramdeni wre Tu samkuTxedi miviReT) 

arsad araa saWiro! 

---------------------------------------------------------------------------------- 

JJJ.3/! jxzfcb! ebkhvgfcjt! )lmbtjgjlbdjjt*! Ufnb. sagnebi 

jgufdeba raime gamaerTianebeli, ganmazogadebeli niSanTvisebis 

mixedviT. es logikis sakiTxebia, uSualod ukavSirdeba 

geometriasac nakvTebis dajgufebas. termins `dajgufeba~ ver 

gaveqceviT. mas masw.-i ixmars ganmartebis gareSe, magram yovelTvis 

mxolod konkretuli amocanis viTarebaSi. moswavleebi TandaTan 

gaigeben, Tu ra moeTxovebaT amgvar amocanebSi. am terminisagan 

mets arc arafers moveliT. sxvaTa Soris, SemdgomSi aseve 

moveqceviT terminebs: `SesaZlebloba~ da `kanonzomiereba~ 

erTerTi yvelaze advilia dajgufeba feris mixedviT (roca 

dasajgufebeli sagnebidan TviTeuli TiTo ferisaa da es ferebi 

mkveTrad gansxvavdeba, magaliTad: wiTlebi, Savebi, mwvaneebi da 

sxva). wminda da mkveTri feri Zalian TvalSisacemi niSanTvisebaa. 

sakmaod Znelia feridan ganyeneba, anu feris ugulebelyofa da 

sxva niSanTvisebis mixedviT dajgufeba, magaliTad, formis mixedviT. 

kidev ufro Znelia logikuri mniSvnelobis mixedviT dajgufeba: 

aveji, mcenareebi, samuSao iaraRebi da sxva. 

erTi kerZo SeniSvna biologiidan: cneba `cxoveli~ aerTianebs 

oTxfexa nadirebsac, qvewarmavlebasac, Tevzebsac da frinvelebisac, 

sazogadod, yvela suliers adamianis garda. ese igi, qaTami Tu 

beRura frinvelebi ki arian, magram cxovelebic arian! 

---------------------------------------------------------------------------------- 

32/4/!ojwUfcjt!ebkhvgfcb!ps!khvgbe!fsUj!ojTboUwjtfcjt!nj. 

yfewjU/! ufsnjoj! èbkhvgfcb~/! dafa-magidaze areulad awyvia: 

ramdenime fanqari, rveuli, wigni, saxazavi, kalmistari; agreTve 

Tojina, saTamaSo manqana, ramdenime patara feradi burTi da ramdenime 

sxva saTamaSo (saTamaSoebis raobas ara aqvs mniSvneloba). 

_ aq sxvadasxva nivTebi awyvia. isini unda gavyoT or 

jgufad. unda davajgufoT or jgufad. aqeT mxares 

gadmovawyoT erTi jgufi, iqiT mxares gadavawyoT meore 

jgufi. Tqveni azriT, rogor jobia am nivTebis dajgufeba?

- 101 - 

gamoiTqmeba sxvadasxva azri. gamova moswavle da cdis 

imgvarad dajgufebas. klasi darwmundeba, rom amgvari dajgufeba 

uazro gamodis. masw.-i SekiTxvebiTa da miTiTebebiT klass 

im azramde miiyvans, rom azriani da kargi dajgufeba _ esaa 

mocemuli nivTebis dayofa saswavlo nivTebad da saTamaSoebad. 

_ maS, erTi jgufi unda iyos saswavlo nivTebi. is, rac 

gvWirdeba swavlisaTvis. aba, naTela, aqeT gadmoawyve saswavlo 

nivTebi! ra dagvrCa? {saTamaSoebi} diax, saTamaSoebi iqiT 

gadavawyoT. ramdeni jgufi miviReT? {ori} radara jgufebia? 

{saswavlo nivTebi da saTamaSoebi}. 

_ axla davakvirdeT suraTs (21.3). ra xatia aq? {xeebi} es 

xeebic or jgufad unda davajgufoT. vin moisazrebs, radara 

jgufebad davajgufoT xeebi? 

gamoiTqmeba sxvadasxva azri. klasi darwmundeba, rom amgvari 

dajgufeba uazro gamovidoda. masw.-i SekiTxvebiTa da 

miTiTebebiT klass im azramde miiyvans, rom azriani da kargi 

dajgufeba _ esaa xeebis dayofa maRal da dabal xeebad. 

_ maS, erTi jgufi unda iyos maRali xeebi. magram rogor 

davajgufoT? xom ver aviRebT da cal kuTxeSi xom ver 

mivwevT! aba ra vqnaT? SeiZleba ase davajgufoT: aarCieT yvela 

maRali xe da yvela erTmaneTTan SeaerTeT mwvane xazebiT! 

(viwyebT erTi ganapira maRali xidan da Tanmimdevrulad 

SevaerTebT yvelas). monakveTebiT SeerTeba araa saWiro, iyos 

uswormasworo xazebi. ra dagvrCa? {dabali xeebi} dabali xeebi 

yvelani erTmaneTTan SeaerTeT wiTeli xazebiT! ramdeni jgufi 

miviReT? {ori} radara jgufebi? 

32/5/!`txsbgj!qbtvyfcj!`!lblvo.ubTfcj~ _ ix. 18.4. 

g 22.!hbnfpsfcb.!tbz/!bnpd/!nplmfcb{f/!ebkhvgfcb!ps!khvgbe/! 

33/1/!21.8 _ SemoqmedebiTia! 

33/2/ Ubwj! 4/2.jt! Tfkbnfcb! eb! hbnfpsfcb. jer Seikribeba 

nivTebis didi grovebi (Sedegis dauTvlelad). xan vityviT 

`Sekribe~, xan _ `miumate~. Semdeg Seikribeba ramdenime 

`dominoseburad~ ganlagebuli magniti: sameuli da wyvili,

- 102 - 

wyvili da oTxeuli da sxva. Sedegi daiTvleba. 

Semdeg moswavleebi gadaSlian mTw-s da misi daxmarebiT 

miumateben ricxvebs. erTi moswavle ityvis Sekrebis erT 

magaliTs, romelic mTw-s mocemul gverdzea, sxva gascems 

pasuxs da ase Semdeg. Semdeg gadaSlian mTw-s sxva gverds da 

ase Semdeg. oRond, am dros masw.-ic CaurTavs SekiTxvebs. 

magaliTad, Tuki moswavlem Tqva magaliTi `sams mivumatoT 

ori~ da meorem gasca pasuxi `xuTi~, masw.-i am dros hkiTxavs 

masve: _ aba, Cqara miTxari, ramdenia ors mivumatoT sami? 

(moswavlem mTw-sa da angariSis gareSe unda Tqvas pasuxi, 

radganac weRan iTqva `sams mivumatoT ori~). 

daskvna: sul erTia, rogor SevkrebT: xuTs mivumatoT erTs 

Tu erTs mivumatebT xuTs (da ase Semdeg). 

bolos, suraTSi 22.1 Seavseben cariel elifsebs. Sedegi 

Caixateba saTanado nakvTSi: eqvseuli _ eqvskuTxedSi da 

xuTeuli _ xuTkuTxedSi. 

33/3/!vnbsujwftj!tbzpgbdypwsfcp!bnpdbofcj!sbpefopcjt!npl. 

mfcb{f/! jer swrafad gakeTdeba ramdenime savarjiSo “-iT 

metis~ dawyobaze (ix. 21.2). _ daawyve xuTze oriT meti kubi! 

dafaze daxate samze samiT meti monakveTi!.. 

_ moismineT amocana. naTelam skolaSi oTxi fanqari waiRo, 

Sin ori fanqari datova. ramdeni fanqari hqonia naTelas? 

umjobesia amoixsnas TvalsaCinoebis gareSe. 

_ axla am amocanis gagrZeleba. maS, naTelas eqvsi fanqari 

hqonia. (dafaze daixazeba eqvsi mokle monakveTi). mere gavada 

dro da naTelas sami fanqari gauTavda. moemata Tu moaklda na- 

Telas fanqrebi? {moaklda} ramdeni fanqari darCenia naTelas? 

kvlav gavameorebinebT pirobas. Semdeg romelime moswavlem 

amoxsnas, dafasTanac gamovides da daxazuli eqvsi monakveTidan 

sami waSalos. sruli pasuxi sxvam Camoayalibos. 

_ moismineT axali amocana. ezoSi oTxi knuti TamaSobda. 

maTgan sami sardafSi daimala. ezoSi knutebis raodenoba 

moimatebda Tu moiklebda? {moiklebda} ramdeni knuti 

darCenila ezoSi? (Cveulebriv amoixsneba).

- 103 - 

_ moswavles saSinao davalebad oTxi amocana hqonda. man 

daamTavra ori amocana da Seisvena. ramdeni amocana darCenia 

mas? (Cveulebriv amoixsneba). 

_ axla daxedeT suraTs (22.2). ai amdeni vaSli iyo TasSi. 

ramdeni yofila? {xuTi} sruli pasuxi! {TasSi xuTi vaSli 

iyo} bavSvma ori vaSli SeWama. moimatebda Tu moiklebda TasSi 

vaSlebis raodenoba? {moiklebda} gadaxazeT imdeni vaSli, 

ramdenic SeiWama (gamoikveTeba sityvebi ìmdeni... ramdenic~)! 

ramdeni vaSli darCeboda TasSi? vin gaimeorebs pirobas? ra 

moqmedeba unda SevasruloT? {xuTs gamovakloT ori} risi 

tolia? {samisa} CamoayalibeT sruli pasuxi {TasSi darCenila 

sami vaSli} gaaferadeT imdeni vaSli, ramdenic TasSi 

darCenila (kvlav gamoikveTeba sityvebi ìmdeni... ramdenic~)! 

_ vin gaixsenebs, ris Sesaxeb iyo pirveli amocana? meore? 

... dRevandel amocanebSi raodenobas emateboda Tu akldeboda? 

{akldeboda} romeli moqmedeba iyo saWiro am amocanebis 

amosaxsnelad? {gamokleba}. 

_ vinc SeZlebs, Sin moigonos kidev aseTi axali amocana. 

gamoklebaze! xval mogviyves. 




mosalodnelia, rom amjerad meti moswavle Seasrulebs am 

sazepiro davalebas. 

33/4/! obybufcjt! ebkhvgfcb! ps! khvgbe! fsUj! ojTboUwjtfcjt! 

njyfewjU!)ojTboUwjtfcjt!TfdwmjU*/!_ Cveni klasis moswavleebi 

rogor SegviZlia davajgufoT or jgufad? {erTi jgufi _ 

gogonebi, meore _ biWebi}. kidev sxvagvaradac SegviZlia 

davajgufoT. magaliTad, ase: erT jgufSi iyos yvela is bavSvi, 

romelic zis ai am adgilze ufro win. xolo meore jgufSi 

vin moxvdeba? {vinc zis am adgilze ufro ukan}. 

_ axla CamoTvaleT, ra xatia am CarCoSi (22.3) {mtredi, 

beli, telefoni...}. radara jgufebad SeiZleba davajgufoT? 

{SeiZleba ase: wiTlebi da ruxebi} kidev sxvagvaradac SeiZleba

- 104 - 

davajgufoT. rogor? {cxovelebi da saojaxo nivTebi} asec 

SeiZleba da isec. Cven avirCioT meore. erT jgufSi iyos yvela 

cxoveli. yvela cxovelis naxati SeaerTeT mwvane xazebiT! 

daimaxsovreT, rom Citic cxovelia! 

meore CarCoSi ras vxedavT? {wiTeli samkuTxedi, ruxi 

samkuTxedi, wiTeli kvadrati, ruxi oTxkuTxedi...} ese igi, 

sul ramdeni samkuTxedia? {oTxi} kvadrati? {ori} sxva oTxkuTxedebi 

ramdenia? {sami} sul oTxkuTxedi ramdenia? {xuTi} 

am nakvTebis dajgufebac orgvarad SeiZleba. aba, rogor da rogor? 

{pirveli _ wiTlebi da ruxebi; meore _ samkuTxedebi da 

oTxkuTxedebi}. avirCioT meore dajgufeba. daajgufeT! ... 

33/5/! `txsbgj! qbtvyfcj~/ masw.-i daaxloebiT 1 wamis 

ganmavlobaSi aCvenebs TiTebs: 2, 3, 4, 5 an 6 TiTs, Tanac, 

iyenebs orive xelis TiTebs. magaliTad, aCevnebs 2 da 3 TiTs. 

moswavlem unda Tqvas, sul ramdeni iyo {xuTi}. meore wreze 

Cvenebis dro kidev ufro Semcirdeba, mesameze _ kidev. icvleba 

TiTebis raodenobac, TviT TiTebic da TiTebis daxrilobac 

(xan gverdiTaa gaSverili, xan _ qveviT, xan _ zeviT, xan _ 

cerad da sxva). 

g 23.!nprnfefcb!hbnplmfcb/!bnpd/!hbnplmfcb{f/!oblwUfcjt!ebkh/ 

---------------------------------------------------------------------------------- 

J.5/! jxzfcb! nprnfefcb! hbnplmfcb; zepiri gamokleba eqvsis 

farglebSi, nivTebiT gamoklebis niSnisa da cifrebis gareSe. 

ZiriTadi sayrdenia dominosebri TvalsaCinoeba. iqneba eqvsis 

farglebSi ZiriTad sxvaobaTa angariSi da pirveli damaxsovreba. 

mTavaria, rom moswavles Seeqmnes safuZvliani da mtkice 

warmodgena imisa, Tu ra aris moqmedeba gamokleba. es ufro 

Znelia, vidre Sekrebis warmodgena. saqme isaa, rom Sekrebis 

gaTvalsaCi-noebisas orive Sesakrebi grovaa mocemuli da maT 

vaerTebT. yvelaferi uSualod Cans. gamoklebis marTebuli 

gaTvalsaCinoebisas ki mocemulia mxolod saklebi. maklebi ar Cans, 

igi mxolod gonebaSia da es qmnis siZneles. 

Tumca, Zalian garcelebulia maklebis gaTvalsaCinobac. magaliTad, 

xatia 6 vaSli da gverdiT uxatia 2 vaSli, rac viTom un-

- 105 - 

da gamoakldes 6 vaSls. magram es Zalian mcdaria, vinaidan gamoklebisas 

saklebs akldeba Tavisive nawili, da ara sxva ram! zogjer 

daxataven 6 vaSls da Semoxazaven 2 vaSls, rac viTom maklebis 

gaTvalsaCinoebaa. es gacilebiT ukeTesia, vidre wina gaTvalsaCinoeba, 

magram mTlad kargi arc igia. saqme isaa, rom gamoklebisaTvis 

ara aqvs mniSvneloba imas, Tu romel ors gamovaklebT. ori vaSlis 

Semoxazvisas ki TiTqos swored am ors gamovyofT. 

Cveneul meTodikaSi mivyvebiT namdvil da bunebriv viTarebas: 

vaTvalsaCinoebT mxolod saklebs, xolo maklebi gonebaSi iyos. es 

siZnele sakmaod advilad gadailaxeba, radgan moswavleebs ukve 

kargad eqnebaT Camoyalibebuli raodenobis warmodgena. mocemulia 

sagnebis mravalricxovani grova da mas vaklebT (movacilebT, 

movaklebT, gamovaklebT) ramdenime sagans. cxadia, am dros 

araviTar SemTxvevaSi ar unda davTvaloT Sedegi es sxva sakiTxia 

da iwyeba ori gakveTilis Semdeg! 

yvelaferi xdeba cifrul CanawerTa gareSe Canawerebi aq 

mxolod masw.-isTvisaa! 

enobrivi SeniSvna. xSiria mcdari warmoTqma: me gamovakle, 

Tqven gamoakleT da sxva. unda iTqvas: me gamovakeli, Tqven gamoakeliT 

da ase Semdeg. Tumca: man gamoaklo, me gamovakleb... 

---------------------------------------------------------------------------------- 

34/2/! hbnplmfcb-! wjUbsd! nprnfefcb! `! ojwUUb! hspwfcjU/! 

`hbnpblfmj~/! momzadebulia ramdenime sxvadasxva grova, TviTeulSi 

12-20 sagniT. dafazec xatia 16 sxvadasxvagvari oTxkuTxedi 

da calke _ 20 sxvadasxva sigrZisa da daxrilobis mqone 

monakveTi. masw.-i rig-rigobiT gamoiZaxebs moswavleebs da daavalebs: 

_ kubebis grovas moakeli, moacile sami kubi! {aiRebs 

sam kubs da Caabrunebs maT Sesanax kolofSi} Cxirebis grovas 

moakeli, moacile xuTi Cxiri! {aiRebs xuT Cxirs da Seinaxavs} 

nuSebis grovas moakeli oTxi nuSi! samkuTxedebis grovas moakeli 

eqvsi samkuTxedi! oTxkuTxedebis grovas moakeli oTxi 

oTxkuTxedi! {waSlis oTxs} monakveTebis grovas moakeli erTi 

monakveTi! {waSlis erTs}. 

_ esaa gamokleba. axla daxedeT suraTs (23.1). risi grovaa

- 106 - 

aq? {TeTri wreebisa} am grovas gamoakeliT eqvsi wre! rogor 

unda gamovakloT? {unda wavSaloT} rogor wavSaloT? 

{wiTlad gadavaferadoT!} gamoakeliT eqvsi! 

34/3/! vnbsujwftj! tbzpgbdypwsfcp! bnpdbofcj! hbnplmfcb{f! `! 

tywbpcjt!np[fcob/!!ufsnjoj!èbobsDfoj~/!jer swrafad gakeTdeba 

ramdenime savarjiSo `-iT metis~ dawyobaze (ix. 21.2). _ 

daawyve xuTze samiT meti kubi! dafaze daxate samze oTxiT 

meti monakveTi!.. 

_ moismineT amocana. naxeT suraTze (23.2) Citebi. amdeni 

Citi ijda xeze. ramdeni Citia? {eqvsi} Semdeg oTxi Citi 

gafrinda. ramdeni Citi darCenila xeze? 

Cveulebrivad amoixsneba (gadaixazeba oTxi Citi) da Semdeg 

suraTze gaferaddeba imdeni Citi, ramdenic darCenila xeze. 

_ meore suraTze ra xatia? {eqvsi soko} diax, mdeloze 

eqvsi soko iyo. maTgan sami soko iyo yviTeli. gaaferadeT! 

danarCeni (am sityvis gamokveTiT) sokoebi ki yavisferi iyo. 

ramdeni soko yofila yavisferi? 

Cveulebrivad amoixsneba da Semdeg suraTze gaferaddeba 

danarCeni (am sityvis gamokveTiT), yavisferi sokoebic. 

_ ojaxSi xuTi bavSvia, maTgan oTxi gogonaa. ramdeni 

biWuna yofila ojaxSi? 

sasurvelia, amoixsnas TvalsaCinoebis gareSe. visac 

gauWirdeba, gamovides dafasTan da daxatos sqematurad xuTi 

kacuna (òjaxSi bavSvebi~), maTgan oTxs dauxatos `kaba~ 

(gogonebi). _ danarCeni, biWuna, ramdenia? 

_ vin gaixsenebs, ris Sesaxeb iyo pirveli amocana? meore? 

... romeli moqmedeba iyo saWiro am amocanebis amosaxsnelad? 

{gamokleba}. 

_ vinc SeZlebs, Sin moigonos kidev aseTi axali amocana. es 

amocana unda iyos imaze, Tu danarCeni (am sityvis gamokveTiT) 

ramdenia. rogorc sokoebze iyo. vinc moigonebs, xval 

mogviyves. 


da SeZlebs, momdevno gakveTilze masw.-i mas Seaqebs da

- 107 - 

wamaxalisebel niSans dauwers.! 

34/4/!obtxbwmj!oblwUfcjt!ebkhvgfcb!ps.ps!khvgbe/!magnituri 

(an lursmnebiani) dafa swori xaziT gayofilia marjvena da 

marcxena mxareebad. cali mxare carielia, xolo meore mxares 

areulad mikrulia sxvadasxva feris, zomebisa da formis: samkuTxedebi, 

oTxkuTxedebi (maT Soris ori kvadrati), xuTkuTxedebi, 

eqvskuTxedebi, wreebi, rgolebi. garda amisa, daxazulia 

ramdenime monakveTi da dasmulia ramdenime wertili. 

_ davajgufoT es nakvTebi. jer gamovides dali da aqeT 

mxares calke daawyos samkuTxedebi. (da ase Semdeg _ yvela 

saxis nakvTi cal-calke jgufebad daewyoba dafis cariel 

mxares. pirvel mxares ki darCeba mxolod monakveTebi da 

wertilebi). ra dagvrCa? {wertilebi da monakveTebi}. maS 

radara jgufebad dagvijgufebia es nakvTebi? pirvel jgufSi 

romeli nakvTebia? {wertilebi da monakveTebi} meore jgufSi? 

{samkuTxedebi, oTxkuTxedebi, xuTkuTxedebi, eqvskuTxedebi, 

wreebi, rgolebi}. 

_ axla daakvirdiT suraTs (23.3). nakvTebi sxvadasxva 

ferisaa! davajgufoT es nakvTebi or jgufad. erT jgufSi iyos 

wreebi da ... ra? ra jobia iyos wreebTan erTad? {rgolebi} 

meore jgufSi ra iqneba? {samkuTxedebi da oTxkuTxedebi} 

daajgufeT, SeaerTeT xazebiT! pirveli jgufis nakvTebi 

SeaerTeT wiTeli xazebiT, meore jgufisa _ Savi xazebiT. 

34/5/!Ùwbmjt!ebybnybnfcb~ -ix. 15.4. 

34/9-Si saWiroa gameoreba: ras niSnavs sityva `garda~ 

{pirvelSi ar unda CavxatoT, yvela sxvaSi, yvela danarCenSi ki 

unda CavxatoT}. garda amisa, masw.-ma moswavleebs urCios, rom 

am amocanis amoxsnisas rveuli mosaxerxeblad moabrunon, raTa 

advilad dainaxon nakvTebi. 

g 24.!hbnplmfcb!hspwfc{f/!UwbmtbD/!`.jU!oblmfcpcb~.!ebkhvg/! 


35/2/!hbnplmfcjt!Tftsvmfcb!ojwUUb!hspwfc{f/!

- 108 - 


35/3/!UwbmtbDjope!npdfnvm!sbpefopcbUb!`.jU!oblmfcpcb~!`! 

nbtx/.jt!njfs!DwfofcjU!eb!nptxbwmjt!njfs!ebxzpcjU/! 

jer swrafad gakeTdeba ramdenime savarjiSo ‘-iT metis~ 

dawyobaze (ix. 21.2). _ daawyve eqvsze oriT meti kubi! dafaze 

daxate erTze samiT meti monakveTi!.. 

Semdeg masw.-i dafamagidaze dadebs 6 kubs. _ vis SeuZlia 

dados eqvsze oriT naklebi raodenobis kubebi? 

masw.-i aclis bavSvebs, mixvdnen, xuTze oriT naklebi raodenobis 

kubebi rogor dadon, mousmens maT azrs. Semdeg axsnis: 

_ magidaze unda davdoT eqvsi kubi, imdenive kubi, ramdenic 

magidazea da Semdeg aviRoT ori kubi. movakloT ori kubi. 

rig-rigobiT gamoiyvans moswavleebs da daawyobinebs raimeze 

oriT naklebi raodenobis Cxirebs, kvadratebs, wreebsa da sxva. 

aseve moviqceviT samiT, oTxiT, xuTiT naklebi nivTebis 

dadebisas. cxadia, Sedegis daTvla araa saWiro. 

_ vinc SeZlebs, Sin moigonos axali amocana, romlis amosaxsnelad 

saWiro iqneba gamokleba. vinc moigonebs, xval 

mogviyves. 




35/4/! ojwUfcjt! ebkhvgfcb! sbnefojnf! khvgbe! fsUj! ojTboUwj. 

tfcjt!njyfewjU-!UjUp!xfwsjt!DbnbufcjU/!!ufsnjoj!`sbnefojnf~/! 

_ aba, vin dadebs magidaze ramdenime (am sityvis 

gamokveTiT) kubs? (gamova moswavle da aiRebs kubebs raTa 

dados}. nika, ramdeni kubi unda dados ciram? {ramdenime} ki, 

marTalia, magram mainc ramdeni cali kubi dados? {SeiZleba 

sami, oTxi, xuTi an eqvsi} cira, daawyve! axla me davdeb 

girCebs (dadebs or girCas). aris Tu ara es ramdenime girCa? 

{ara, ori ar aris ramdenime, ori Zalian cotaa} erTi girCa 

rom damedo? {erTi kidev ufro cotaa, arc isaa ramdenime} 

axla Tamari gamovides da dafaze daxatos ramdenime oTxkuTxedi, 

naTiam ki ramdenjerme Semokras taSi! kargia. axla

- 109 - 

sandrom ramdenjerme daakakunos. axla daxedeT, am kolofSi ra 

aris? (kolofSi ocdaaTiode an meti magniti an Rili yria). 

aris Tu ara es ramdenime magniti? {ara, magnitebi Zalian 

bevria da amitom araa ramdenime} diax, aq mravali (am sityvis 

gamokveTiT), Zalian bevri magnitia. ramdenime iqneboda, ufro 

cota rom yofiliyo: sami, oTxi, xuTi an eqvsi. gamovides akaki 

da xelSi daikavos mxolod ramdenime magniti! 

Semdeg dafaze swrafad keTdeba igive, rac 23.3-Si. oRond, 

Tavidanve amocanaa ara or, aramed ramdenime jgufad dajgufeba. 

dajgufebis Semdeg TviTeuli jgufi (eqvskuTxedebi, samkuTxedebi, 

wreebi, rgolebi, monakveTebi da sxva) cal-calke 

Semoixazeba. 

_ feriT Tu hgavs es nakvTebi erTmaneTs? (masw.-i TiTiT 

erT jgufs Semoxazavs) {ara, feri sxvadasxva aqvT} feriT ar 

hgavs, magram mainc erT jgufSia! ratom? {imitom, rom aq 

yvela wreebia, aq yvela samkuTxedebia... }. 

_ axla davaTvalieroT es suraTi (24.3). es naxatebic 

ramdenime jgufad unda davajgufoT. rasTan davajgufoT Tevzi? 

{ixvTan} ratom? {orive curavs da orive cxovelia} kidev 

Tu aris sxva cxoveli, maTTan erTad rom davajgufoT? {ara, 

meti cxoveli araa} dasajgufeblad SegviZlia naxatebi xazebiT 

SevaerToT. magram ramdenime jgufi gvaqvs da xazebi Zalian 

gadaixlarTeba. amitom modi, jgufebi cal-calke SemovxazoT. 

cxovelebis naxatebi SemovxazoT mwvane xaziT (Tan dafaze 

kidev erTxel axsenebs, ras niSnavs `Semoxazva~). doqi rasTan 

davajgufoT? {larnakTan} maTTan kidev ra davajgufoT? {Tasi} 

ratom? {samive WurWelia} kidev Tu aris maTTan rame dasajgufebeli 

{ara}... WurWeli ra ferad SemovxazoT? kargi, iyos 

lurji. ... (da ase dajgufdeba iaraRebi, Semdeg mcenareebi). 

maS, ramden jgufad davajgufeT es naxatebi? {oTx jgufad}. 

35/5/! `txsbgj! qbtvyfcj~/ masw.-i TviTeul moswavles rigrigobiT 

usaxelebs rames, man ki maSinve unda gamoicnos: erTia, 

oria, ramdenimea Tu mravalia. (unda iTqvas mxolod am oTxi 

pasuxidan erTerTi). Tanac, moswavlem pasuxi sityviT ki ar

- 110 - 

unda Tqvas, aramed: erTze _ awios erTi TiTi, orze _ ori 

TiTi, ramdenimeze _ ramdenime TiTi, mravalze _ orive xeli 

orive gaSlili xelisguliT (mravali TiTi). am TamaSSi sul 

ixmareba sityva `mravali~ (da ara `bevri~). 

dasaxelebulTa nimuSebi: xelisa da fexis TiTebi erTad; 

vefxvis Tvalebi; Jirafis kiseri; spilos xorTumi (Zalian 

kargi SekiTxvebia _ didisa da mravlis gasarCevad!); xoWos 

fexebi; cal xelze TiTebi; manqanis borblebi; qoris frTebi; 

ZaRlis fexebi; Cveni skolis ezoSi mdgari xeebi; Sens saxlSi 

kovzebi; axla quCaSi mimavali adamianebi... 

g 25.!hbnplmfcb/ `.jU!oblmfcpcb~.!ebkhvg/!sbnefojnf!khvgbe/! 

36/1/ dawvrilebiT unda gairCes 24.6. 

_ aq davajgufeT erTmaneTs msgavsi nakvTebis wyvilebi. 

msgavsi, ese igi, romelsac hgavs yvelaze metad. aseTi wyvilebi 

erTmaneTTan xazebiT SevaerTeT. rasTan SevaerTeT zeda ruxi 

oTxkuTxedi? {vardisfer oTxkuTxedTan} ruxi wre? {wiTel 

wresTan} Savi nakvTi ra aris? {kvadrati} rasTan SevaerTeT? 

{wiTel kvadratTan} wiTeli samkuTxedi? {Sav samkuTxedTan} 

ruxi rgoli? {meore rux rgolTan} ramdeni jgufi miviReT? 

{xuTi} diax, es nakvTebi davajgufeT xuT jgufad. TiTo 

jgufSi ramdeni nakvTia? {ori} diax, TiTo jgufSi or-ori 

nakvTia, erTmaneTis msgavsi nakvTebi. feriT Tu hgavs es 

nakvTebi erTmaneTs? {ara, feri sxvadasxva aqvT} feriT ar 

hgavs, magram mainc msgavsebia. ratom? {imitom, rom erTic wrea 

da meorec, erTic samkuTxedia da meorec... }. 

36/2/!bewjmj!hbnplmfcb!7.jt!gbshmfcTj;!3`2-!4`2-!4`3/! 

Tavidan kidev erTxel gavimeorebT gamoklebas grovebze, 

rogorc iyo 23.1-Si, gavimeorebT swrafad. 

dafazea `dominoseburad~ ganlagebuli oreuli, sameuli da 

sameuli. _ gamovides malxazi da gamoaklos ors erTi, 

danarCenma moswavleebma ki daxedeT aseTsave suraTs (25.1) da 

suraTze gamoakeliT ors erTi. ra miviReT? {erTi} diax, ors 

gamoklebuli erTi tolia erTisa. gaimeoros vaJam!

- 111 - 

aseve Sesruldeba gamokleba 3_1 da 3_2. moqmedebaTa Sesruleba 

yvela SemTxvevaSi saWiroa, bavSvebma romc zepiradve 

Tqvan sxvaobis mniSvneloba, mainc. am SemTxvevaSi masw.-i ikiTxavs: 

_ saidan ici, rom sams gamoklebuli ori erTis tolia? 

gviCvene! (da moswavle iZulebuli gaxdeba, Caataros moqmedeba). 

Semdeg mTw gadaiSleba `3~-ze. _ rogoraa aq dawyobili 

sameuli? {ori da erTi}. ori da erTi. erTi rom gamovakloT, 

ra dagvrCeba? aba daafareT TiTi erTs, ra darCa? {ori} diax, 

sams gamoklebuli erTi tolia orisa! ori rom gamovakloT, 

ra dagvrCeba? daafareT TiTi ors, ra darCa? {erTi} diax, sams 

gamoklebuli ori tolia erTisa! 

36/3/! sbpefopcbUb! `.jU! oblmfcpcb~! `! nptxbwmjt! njfs! 

ebxzpcb/! ! ! ! jer swrafad gakeTdeba ramdenime savarjiSo ~-iT 

naklebis~ dawyobaze (ix. 24.2). _ ai, me davawyve xuTi kubi. 

Sen daawyob xuTze samiT nakleb kubs! ai, me davxate oTxi wre. 

Sen daxatav oTxze samiT nakleb wres!.. 

yvelaferi iseve keTdeba, rogorc 21.2-Si, oRond metoba 

icvleba naklebobiT. 

bolos, cariel CarCoSi (25.2) Caixateba eqvsze oriT 

naklebi raodenobis sxvadasxvagvari oTxkuTxedebi. 

_ Sin yvelam unda moigonos axali amocana, romlis amosaxsnelad 

saWiro iqneba an Sekreba, an gamokleba. es davaleba 

ukve yvelasTvisaa! 

es ar iqneba savaldebulo davaleba. vinc karg amocanebs 

Seadgens, momdevno gakveTilze masw.-i mas Seaqebs da 


36/4/! obybufcjt! ebkhvgfcb! sbnefojnf! khvgbe! fsUj! ojTbo. 

Uwjtfcjt! njyfewjU/! dafaze keTdeba igive, rac 24.3-Si. aqac 

Tavidanve amocanaa ramdenime jgufad dajgufeba. dajgufebis 

Semdeg TviTeuli jgufi cal-calke Semoixazeba. garda amisa, 

bolos emateba erTi davalebac: _ axla yvela jgufs unda 

CavumatoT kidev erTi nakvTi. TiTo jgufSi unda CavamatoT 

TiTo nakvTi (sityvebis `TiTo~ gamokveTiT). ramdenia TiTo? 

{erTi} diax, TiTo jgufSi erTi nakvTi. gamovides naTela. ra

- 112 - 

nakvTi CavumatoT samkuTxedebis jgufSi? {samkuTxedi} Caumate! 

(da ase Semdeg _ sxvadasxva moswavleebi rig-rigobiT 

Caumateben TiTo imdagvarive nakvTs). 

_ axla davaTvalieroT suraTi (25.3). es naxatebic ramdenime 

jgufad unda davajgufoT. radara jgufebis naxatebs xedavT? 

masw.-i SekiTxvebiTa da misaxvedri miTiTebebiT klass gamoayofinebs 

Semdeg jgufebs: xil-bostneuli, saojaxo nivTebi, mwerebi, 

saswavlo nivTebi. SeiZleba xili da bostneuli cal-calkec iyos 

_ nu SevusworebT; SeiZleba, erTad iyos da uwodon ara `xilbostneuli~, 

aramed `saWmelebi~ _ nurc amas SevusworebT. 

_ dasajgufeblad SegviZlia naxatebi xazebiT SevaerToT. 

magram ramdenime jgufi gvaqvs da xazebi Zalian gadaixlarTeba. 

amitom modi, jgufebi cal-calke SemovxazoT. xil-bostneulis 

naxatebi SemovxazoT mwvane xaziT. saojaxo nivTebi ra ferad 

SemovxazoT? kargi, iyos lurji. mwerebi ra ferad 

SemovxazoT? kargi, wiTlad. ra dagvrCa? {saswavlo nivTebi} 

eseni ra ferad SemovxazoT? kargi, iyos Savi. 

_ axla TiTo jgufSi unda CaumatoT kidev erTi naxati. ra 

CavxatoT xil-bostneulis jgufSi? kargi, CavxatoT kitri! (da 

ase Semdeg _ yovel jgufs TiTo saTanado naxati Caemateba). 

36/5/! `txsbgj! qbtvyfcj! `! tbqjsjtqjspoj~/ masw.-i ambobs 

romelime sityvas naswavl raodenobriv da sivrciT mimar- 

TebaTagan: bevri, mravali, cota, didi, patara, maRali, dabali, 

zemo, zemoT, zeda, qvemo, qvemoT, qveda, wina, win, ukana, ukan, 

SigniT, Siga, gareT, gare, msxvili, wvrili, sqeli, Txeli, 

farTo, ganieri, viwro. 


amaTi sapirispiroebi Znelia. amitom eseni meore wreze unda 





g 26.!hbnplmfcb/!hbnfpsfcb/!oblwUfcjt!ebkh/!sbnefojnf!khvgbe/!

- 113 - 

37/1/ dawvrilebiT unda gairCes 25.6. bolos Sejamdes: _ 

maS, ramdeni jgufia? {oTxi} ramdeni nakvTia pirvel jgufSi? 

{oTxi} meoreSi? {imdenive, oTxi} mesame jgufSic imdenivea? 

{ara, samia} meoTxe jgufSi? {ori}. 

25.7-is kargad Sesrulebuli da saintereso naxatebi 

dafazec daixatos. es amocana _ SemoqmedebiTia! 

25.8-Si yuradRebis miqceva sWirdeba zolebis erTmaneTTan 

gadakveTis adgilebs _ kargad Tua gaferadebuli. 

37/2/!hbnplmfcb;!5`2-!5`4-!5`3/! 

am monakveTs SedarebiT cota dro vamyofinoT. 

jer gavimeorebT 25.1-is gamoklebis magaliTebs. Semdeg 

SevasrulebT axal moqmedebebs iseve, rogorc 25.1-Si: jer 

dafaze da Tan msr-Si (suraTi 26.1), Semdeg ki mTw-s `4~-is 

meSveobiT. mTw-s ganxilvisas vekiTxebiT: _ ramdenia oTxs 

gamovakloT erTi? oTxs gamovakloT sami? oTxs gamovakloT 

ori? rogor mixvdiT? {aq oTxeuls erTi rom movakloT, sami 

darCeba. sami rom movakloT, erTi darCeba. ori rom movakloT, 

ori darCeba}. 

Sedegebs aucileblad sityvieradac CamovayalibebinebT srulad. 

TfojTwob;! sasurvelia swored imgvari Tanmimdevroba, rogorc 

miTiTebulia. 4_1 gamoklebas unda mosdevdes ara 4_2, 

aramed 4_3, radganac swored mas ukavSirdeba 4_1 `dominoseburi~ 

TvalsaCinoebiT. 

aseve iqneba gamoklebis sxva SemTxvevbSic, rasac aRar mivuTi- 

TebT. ubralod, sasurvelia miTiTebul TanmimdevrobaTa dacva. 

37/3/! Ubwj! 5/3.jt! hbnfpsfcb! eb! Tfkbnfcb/! ! jer gairCeva 

zepirad Sesasrulebeli saSinao davaleba _ Sesadgeni 

amocanebi. moswavleebi amoxsnian TanaklaselTa mier Sedgenil 

karg amocanebs, romlebsac masw.-i gamoarCevs da Seaqebs. 

Semdeg keTdeba `-iT met-naklebobis~ savarjiSoebi: 

_ daawyve oTxze samiT meti raodenobis girCebi! dafaze 

daxate xuTze xuTiT meti raodenobis oTxkuTxedebi. daxate 

rvaze oTxiT meti raodenobis monakveTebi! (da ase Semdeg). 

bolos, klass miecema igive saSinao davaleba, rac iyo 25.2-Si.

- 114 - 

37/4/!obtxbwmj!oblwUfcjt!ebkhvgfcb!sbnefojnf!khvgbe/!nbhb. 

mjUbe;! xsffcj-! shpmfcj-! lwbesbufcj! eb! tywfcj/! rogorc wina 

gakveTilebze, dajgufdeba nakvTebi. magram dajgufebisaTvis 

gamoviyenebT ferebs. _ jer davajgufoT kvadratebi. magram aq 

imdeni nakvTia, rom ginda xazebiT SevaerToT da ginda 

SemovxazoT, yvelaferi erTmaneTSi gadaixlarTeba! amitom ase 

moviqceT: TiTo jgufi TiTo ferad gavaferadoT! ra ferad 

gavaferadoT kvadratebi? kargi, mwvaned. gaaferadeT yvela 

kvadrati mwvaned, yvela (am sityvis gamokveTiT)! axla 

davajgufoT danarCeni nakvTebi. ra nakvTebs xedavT? {wreebs}. 

ra ferad gavaferadoT? kargi, lurjad. gaaferadeT! kidev ra 

jgufis nakvTebs xedavT? {rgolebs} ra ferad gavaferadoT? ... 

kidev ra jgufis nakvTebs xedavT? {wertilebs} amaT nuRar 

gavferadebT. kidev? ... kidevaa: oTxkuTxedebi, monakveTebi, 

xuTkuTxedebi, eqvskuTxedebi. eseni aRar gaferaddeba. 

_ axla TviTeul gaferadebul jgufs davumatoT kidev 

TiTo nakvTi! romeli nakvTi davumatoT pirvel jgufs? 

{kvadrati} ra ferad davxatoT? {mwvaned} ratom? {yvela 

kvadrati mwvanea da imitom}... da ase Semdeg. 

37/5/! `txsbgj! qbtvyfcj~!`!`.jU! nfupcb~/ masw.-i Semokravs 

taSs ramdenjerme (xan samjer, xan xuTjer...). moswavlem unda 

Semokras taSi: pirvel wreze _ erTiT meti taSi, meore wreze 

_ oriT meti, mesameze _ samiT meti da ase Semdeg. 

masw.-i auxsnis: _ Tqven yuradRebiT mismineT da gonebaSi 

daTvaleT Cemi taSebi. Semdeg SemokariT imdenive taSi, 

ramdenic me, da kidev daumateT erTi taSi. ramdeniT meti taSi 

iqneba Tqveni? {erTiT meti} diax, erTiT meti taSi. 

òriT mets~ jer ar axsenebs, axsenebs pirveli wris 

CamTavrebis Semdeg. 

g 27.!hbnplmfcb/!nbskwojw!/!nbsdyojw/!Tpsjt/!hbnfpsfcb/! 

38/2/!hbnplmfcb;!6`2-!6`5-!6`3-!6`4/! 


SevasrulebT axal moqmedebebs iseve, rogorc 26.1-Si: jer

- 115 - 


meSveobiT. mTw-s ganxilvisas vekiTxebiT: _ ramdenia xuTs 

gamovakloT erTi? xuTs gamovakloT oTxi? ... rogor mixvdiT? 

sasurvelia aseTi pasuxi: _ aq xuTeuls ori rom 

movakloT, sami darCeba (da ase Semdeg). 


--------------------------------------------------------------------------------- 

JJ.5/! jtxbwmfcb! tjwsdjUj! njnbsUfcb! nbskwojw! /! nbsdyojw/! 

Tpsjt/ mas vakavSirebT naswavl rigobiT ricxvebTan, maT Soris 

axlebTan: meSvide, merve, mecxre. 

marjvniv / marcxniv mraval moswavles SeiZleba eSlebodes sakuTar 

TavTan mimarTac ki. miT umetes ufro Znelia am mxareebis 

garkveva sxvis mimarT. am mimarTebas yvela moswavle kargad ver 

daeufleba. 

SemdgomSi sivrciT mimarTebaTa Tema ufro TamaSebSi ganviTardeba. 

---------------------------------------------------------------------------------- 

38/3/! nbskwojw! /! nbsdyojw/! Tpsjt/! `sb! bsjt! bnjt! nbskwojw! /! 

nbsdyojw@~! jer swrafad gakeTdeba ramdenime savarjiSo ‘-iT 

naklebis~ dawyobaze (ix. 25.2). _ daawyve eqvsze oriT 

naklebi kubi! dafaze daxate xuTze oriT naklebi monakveTi!.. 

_ yvelam awios marjvena xeli! dauSviT. axla awieT 

marcxena xeli! miCveneT marcxena yuri! marjvena Tvali! vaJa, 

vin zis Sens marjvniv? Sens marcxniv? Sen visa da vis Soris 

zixar? lali, Sen visa da vis Soris zixar? romel mxares 

gizis levani? fanjrebi Tqvens romel mxaresaa? kari? laSa, 

giorgis marjvniv vin zis? ar SegeSalos, romelia giorgis 

marjvena xeli? giorgis marcxniv? 

am gakveTilze amgvari SekiTxvebi daismis mxolod im 

SemTxvevaSi, roca ori moswavle erTmaneTis mimarT zurgiT 

arian _ asecaa bunebrivad merxebze jdomisas. xolo roca 

meore adamiani bavSvis mimarT pirisaxiTaa _ misi marjvena / 

marcxena piriqiTaa! es Zneli SemTxvevaa da SemdgomSi iqneba. 

sxvis marjvniv / marcxniv mainc Zneli gasarkvevia. amitom 

masw.-i aswavlis:

- 116 - 

_ unda warmoidgino, rom viTom Sen xar imis adgilas. 

romelia Seni marjvena xeli? imis adgilas rom yofiliyavi, 

saiT iqneboda marjvena mxare? midi da Seamowme! 

ase unda gavakeTebinoT yvela moswavles, visac ki 

gauWirdeba warmodgeniT garkveva. 

_ ra aris marcxenas sapirispiro? marjvenasi? 

_ axla davxedoT suraTs (27.2). gogona Cvens mimarT 

zurgiT, zurgSeqceviT dgas. rasa da ras Soris dgas gogona? 

{kedelsa da skams Soris} ra aris gogonas marjvniv? {skami} 

kedeli gogonas romel mxaresaa? {marcxniv} skami romel 

mxaresaa? {marjvniv} gogonas marcxniv ra aris? {kedeli} 

(ramdenime moswavles vekiTxebiT sxvadasxvagvarad). 

Semdeg gadavSliT mTw-s ydis suraTs da davsvamT 

amdagvarsave SekiTxvebs _ oRond, mxolod im cxovelebisaTvis, 

romlebic zurgSeqceviTaa daxatuli! magaliTad: _ vinaa 

bokveris marcxniv? zRarbis romel mxaresaa ku? 


am monakveTs SedarebiT mcire dro eyofa. 

suraTze 27.3 nakvTebi ukve TiTqmis dajgufebulia. 

moswavleebi gaizreben am dajgufebas da TviTeul jgufs calcalke 

Semoxazaven (sasurvelia, sxvadasxva ferebiT). Semdeg 

TviTeul jgufSi Caumateben TiTo nakvTs. bolos jgufebi 

ricxvebs daukavSirdeba. 

_ qvemoT raodenobebia, SeamCnieT? romeli jgufi romel 

raodenobasTan SevaerToT? magaliTad, samkuTxedebi romel 

raodenobasTan SevaerToT? {sameulTan} ratom? {samkuTxeds 

sami gverdi aqvs} danarCenebi ki Sin unda SeaerToT! 

38/5/! `hvebTj! yfmjt! Dbzpgb! eb! hbnpdopcb~/ masw.-i naWris 

gudaSi Cayris plastmasis an magari muyaos kvadratebsa da 

sxvagvar oTxkuTxedebs, wreebs, rgolebs, sxvadasxvagvar 

samkuTxedebs, xuTkuTxedebs, eqvskuTxedebs. _ aqaa kvadratebi, 

wreebi, rgolebi, sxvadasxvagvari samkuTxedebi da kidev sxva 

nakvTebi _ xuTkuTxedebi, eqvskuTxedebi da sxva. Tqven rigrigobiT 

unda CayoT gudaSi xeli, aiRoT erTi nakvTi da

- 117 - 

gudaSi Cauxedavad, xeliT mosinjviT swrafad gamoicnoT, es 

ra nakvTia: kvadrati, wre, rgoli, sxvadasxvagvari samkuTxedi 

Tu sxva rame. xuTkuTxedebis, eqvskuTxedebisa da sxvadasxvagvari 

oTxkuTxedebis gamocnoba Znelia. amitom maT nu gamoicnobT, 

TqviT mxolod: sxvagvari nakvTi. 

pasuxis Semdeg moswavle amoiRebs nakvTs da yvela naxavs, 

SeeSala Tu ara. Semdeg nakvTs isev Caabruneben gudaSi, masw.-i 

aurevs da momdevno moswavles Caayofinebs xels. 

38/8/ saWiroa mcireodeni axsna. dafaze imdagvaradve 

daixateba samkuTxedebi, eqvsiode, da gairkveva, romlebi Cans 

`zemoT~ da romlebi _ qvemoT. 

g 28.!hbnplm/!`.jU!nfu/oblm~/!dysjmjt!tusjr/-!twfu/-!!vksfej/! 

39/1/ 27.8-is garCevisas unda gairkves: _ oTxkuTxedebi 

rasTan SeaerTeT? {oTxeulTan} ratom? {oTxkuTxeds oTxi 

gverdi aqvs} xuTkuTxedebi? {xuTeulTan} ratom? ... 

eqvskuTxedebi? {eqvseulTan} ratom? ... wyvilTan, oreulTan 

raRa SevaerToT? {monakveTebi} ratom? {monakveTs ori bolo 

aqvs} erTeulTan romeli jgufi SevaerToT? {wertilebi}. 

39/2/!hbnplmfcb;!7`2-!7`6-!7`3-!7`5-!7`4/! 


SevasrulebT axal moqmedebebs iseve, rogorc 27.1-Si: jer 


meSveobiT. mTw-s ganxilvisas vekiTxebiT: _ ramdenia eqvss 

gamovakloT erTi? eqvss gamovakloT xuTi? ... rogor mixvdiT? 

sasurvelia aseTi pasuxi: _ aq eqvseuls ori rom 

movakloT, oTxi darCeba (da ase Semdeg). 


39/3/!nbskwojw!/!nbsdyojw/!Tpsjt/!`ft!sjt!nbskwojwbb!/!nbsd. 

yojwbb@~! !`.jU! nfupcb! /! oblmfcpcb~/! jer swrafad gakeTdeba 

ramdenime savarjiSo ‘-iT naklebis~ dawyobaze (ix. 25.2). _ 

daawyve xuTze oTxiT naklebi kubi! dafaze daxate eqvsze samiT 

naklebi monakveTi!.. Semdeg gameordeba imdagvarive SekiTxvebi, 

rogorebic iyo 27.2-Si. Semdeg sakiTxi Zneldeba:

- 118 - 

_ nani, Sen ris marcxniv zixar? vaxo, vis marjvniv zis 

nani? vis marcxniv zis ciala? ris marjvniv kidia es suraTi? 

magram am gakveTilzec vifarglebiT mxolod im SemTxvevebiT, 

roca bavSvebi erTmaneTis mimarT zurgiT arian (raTa 

saWiro ar gaxdes mimarTebis Sebruneba). 

amgvari SekiTxvebi sakmaod daismis. visac gauWirdeba 

garkveva, mivides saTanado adgilas da sakuTari xelebis 

mixedviT gaarkvios. 

Semdeg gadavSliT mTw-s ydis suraTs da davsvamT 

amdagvarsave SekiTxvebs _ oRond, mxolod im cxovelebisaTvis, 

romlebic zurgSeqceviTaa daxatuli! magaliTad: _ melas 

marjvniv vinaa? vis marcxnivaa zRarbi? vis marjvnivaa ciyvi? 

bolos, msr-s 28.2 cariel CarCoSi masw.-is karnaxiT klasi 

Caxatavs: _ CarCos SuaSi daxateT wiTeli monakveTi. mis 

marjvniv daxateT Savi samkuTxedi. wiTeli monakveTis marcxniv 

ki daxateT Savi rvakuTxedi. Semdeg daxateT oTxkuTxedi ise, 

rom wiTeli monakveTi misgan marcxniv iyos. Semdeg daxateT 

wre ise, rom wiTeli monakveTi misgan marjvniv iyos. 

---------------------------------------------------------------------------------- 

JJJ.4/!jxzfcb!dysjmjt!Ufnb/ cxrili ivseba jer yvelaze advili 

da TvalsaCino masaliT. arsebiTad, amisaTvis saWiroa saganTa daxarisxeba 

ori Zalian TvalsaCino niSanTvisebis mixedviT: ferisa da 

formis mixedviT. oRond, am daxarisxebas aadvilebs cxrili. 

Semdeg Semogvaqvs SedarebiT rTuli mravalkuTxedebi. iseve 

rogorc adre, maTi Tanmimdevroba kvlav ariTmetikis xaziTaa 

ganpirobebuli: Svideuli / Svidi / SvidkuTxedi da ase Semdeg. 

---------------------------------------------------------------------------------- 

39/4/!gfsbej!oblwUfcjU!Tfwtfcvmj!dysjmjt!tusjrpoj-!twfuj! 

eb!vksfej/ dafaze xatia carieli cxrili 6 striqoniTa da 

5 svetiT. _ ai, esaa cxrili (masw.-i miuTiTebs dafaze). am 

cxrils aqvs eqvsi striqoni. ai, misi striqonebi (da TiTiT 

gaxazavs TviTeul striqons, Tan iTvlis: erTi, ori, sami, 

oTxi, xuTi, eqvsi). sul eqvsi striqonia. gamovides eka da 

gviCvenos zemodan meore striqoni! arCilma gviCvenos zemodan

- 119 - 

meoTxe striqoni! Teonam _ zemodan pirveli, salomem _ mesame. 

_ ramdeni sveti aqvs am cxrils? {xuTi}. diax, ai svetebi: 

(da TiTiT gaxazavs TviTeul striqons, Tan iTvlis: erTi, 

ori, sami, oTxi, xuTi). sul xuTi stvetia. gamovides irine da 

gviCvenos meore sveti! mananam gviCvenos meoTxe sveti! iraklim 

_ pirveli, leqsom _ mesame. 

svetebi yovelTvis marcxnidan iTvleba. amas Tqma ar 

sWirdeba. masw.-i TiTiT uCvenebs mimarTulebas. 

_ oRond, es cxrili carielia. axla daaTvaliereT es 

suraTi Tqvens rveulSi (28.3). radara xatia am cxrilSi? 

{yvavilebi, manqanebi, burTebi, fisoebi, TviTmfrinavebi} 

daTvaleT Tqvens cxrilSi, ramdeni striqonia? {oTxi} diax, 

oTxi striqonia. moZebneT sul pirveli striqoni, zemodan 

pirveli! ra ferisaa naxatebi? {araa gaferadebuli} moZebneT 

meore striqoni. ra ferisaa naxatebi? {ruxi} mesame 

striqonSi? {wiTeli} meoTxe striqonSi? {Savi}. maS, meramdene 

striqonia wiTlebisa? {mesame} ruxebisa? {meore} Savebis 

striqoni meramdenea? {meoTxe}. 

_ axla vnaxoT svetebi! sul ramdeni svetia? {eqvsi}. mesame 

sveti risia? {manqanebisaa} mexuTe? {fisoebisaa} pirveli? 

{ferebisaa}. yvavilebis sveti meramdenea? {meore}. 

_ axla moZebneT meoTxe striqoni. am striqonSi moZebneT 

mesame ujredi! ra xatia Sig? {Savi manqana}. 

amgvari SekiTxvebi gagrZeldeba TamaSSi (28.4). 

TfojTwob/ cxrilis Sevsebis wesi (striqonisa da svetis 

gadakveTis ujredi...) jer araa saWiro! 

39/5/! `txsbgj! qbtvyfcj! `! dysjmj~/ SekiTxvebi iseTivea, 

rogorebic iyo 28.3-Si. pirvel wreze: _ ra ferisaa mesame 

striqoni? meramdenea ruxi striqoni? risi svetia meeqvse? 

meramdenea TviTmfrinavebis sveti? (da ase Semdeg). 

momdevno wreebze SekiTxvebi rTuldeba: _ ra xatia mesame 

striqonis meoTxe ujredSi? mesame striqonis meoTxe ujredi! 

{wiTeli burTi}. saWiroa iTqvas ferica da saganic. 

g 29.!hbnfpsfcb/!nbsk/nbsdy/!eb!sjhpc/!sjdyw/!!twfujt!Tfwtfcb/!

- 120 - 

3:/1/ 39/7-Si yuradReba gamaxvildeba sityvebze `SigniT~ da 

`gareT~. 

3:/2/!Ubwj!5/2.jt!Tfkbnfcb!eb!hbnfpsfcb/!_ wina gakveTilebze 

ra moqmedeba SeviswavleT? {gamokleba} vin Seasrulebs 

moqmedeba gamoklebas? gamoakeli sami! (uTiTebs grovaze). 

masw.-i dafaze debs eqvs magnits da moswavleebs avalebs, 

saTiTaod gamoaklon erTi, ori, sami, oTxi, xuTi. yoveli 

gamosuli moswavle Sedegs sityvierad ayalibebs: _ eqvs 

gamovakloT erTi darCeba xuTi. _ eqvs gamovakloT ori 

darCeba oTxi. _ eqvs gamovakloT sami darCeba sami. _ eqvs 

gamovakloT oTxi darCeba ori. _ eqvs gamovakloT xuTi 

darCeba erTi. 

aseve viqceviT xuTidan gamoklebaze, oTxidan gamoklebaze, 

samidan gamoklebaze. 

3:/3/! nbskwojw! /! nbsdyojw! njnbsUfcjt! eblbwTjsfcb! sjhpcjU! 

sjdywfcUbo! qjswfmjebo! nffrwtfnef/! gamoviyvanT eqvs moswavles 

da davayenebT erTmaneTis gverdiT. _ vinaa yvelaze marjvniv? 

yvelaze marcxniv? vinaa nunusa da nonas Soris? visa da vis 

Sorisaa leqso? vinaa pirveli? marjvnidan sxvaa da marcxnidan 

_ sxva! marjvnidan pirveli vinaa? marcxnidan pirveli? marcxnidan 

meore? mesame? marjvnidan meoTxe? malxazi meramdenea? 

marjvnidan meramdenea? marcxnidan? (da ase Semdeg). 

Semdeg ganvixilavT suraTs (29.2). daismis amdagvarive SekiTxvebi: 

_ romeli nakvTia yvelaze marjvniv? yvelaze marcxniv? 

wertilsa da samkuTxeds Soris romelia? eqvskuTxedi 

rasa da ras Sorisaa? romelia marjvnidan pirveli nakvTi? 

marcxnidan pirveli? marcxnidan meore? mesame? marjvnidan 

meoTxe? samkuTxedi marjvnidan meramdenea? oTxkuTxedi 

marcxnidan meramdenea? kvadrati? (da ase Semdeg). 

_ xvalisaTvis moigoneT amocana marjvena-marcxenas Sesaxeb! 

3:/4/! gfsbej! obybufcjU! Tfwtfcvmj! dysjmjt! fsUj! twfujt! 

Tfwtfcb/! ! iseve, rogorc 28.3-Si, daiTvleba 29.3 cxrilis 

striqonebi, Semdeg _ svetebi. SekiTxvebic iseTivea: _ meore 

svetSi ra aris? {wignebi} mesameSi? (da ase Semdeg). wiTlebis

- 121 - 

striqoni meramdenea? ra xatia mesame striqonis mexuTe 

ujredSi? ra xatia meoTxe svetis sul qveda ujredSi? ... ... 

_ erTi sveti araa Sevsebuli da Cven unda SevavsoT. meramdene 

svetia Sesavsebi? {meeqvse}. modi, SevavsoT. risi svetia es? 

{kaTxebisa} ra ferisa unda iyos meore striqonis kaTxa? 

{ruxisa} gaaferadeT! mis qveS, mesame striqonisa? {wiTeli} 

gaaferdeT! mis qveS, meoTxe striqonisa? {Savi} gaaferdeT! 

_ Sin aseve SeavsebT, gaaferadebT meore cxrils (29.7)! magram 

am cxrils meoTxe striqonSi feri ara aqvs SerCeuli. modi, 

Cven SevurCioT! romeli feri girCevniaT? kargi, iyos mwvane. 

maS gaaferadeT mwvaned pirveli svetis sul qveda ujredi! es 

ujredi carielia da is unda SeavsoT mwvane feriT. gauferadebel 

naxatebs ki Sin gaaferadebT! 


gadakveTis ujredi...) jer isev araa saWiro. moswavleebi 

gumaniT mixvdebian, rogor Seavson carieli ujredebi Sin. 

3:/5/! `txsbgj! qbtvyfcj~/ SekiTxvebi moicavs Sekreba-gamoklebis 

naswavl magaliTebs: oTxs miumate ori, xuTs gamoakeli 

oTxi da ase Semdeg. 

g 30.!Twjefvmj-!Twjej/!psnbhj!tjwsdjUj!njnbsU/!TwjelvUyfej/ 

---------------------------------------------------------------------------------- 

J.6/!Tfnphwbrwt!sbpefopcboj!Twjejebo!dysbnef/ ricxvis pirveli 

Semotana xdeba iseve, rogorc eqvsamde ricxvebis SemTxvevaSi: 

yvelaferi keTdeba mxolod TvalsaCinod mocemuli nivTebis grovebiT, 

cifrul niSanTa da CanawerTa gareSe. ZiriTadi sayrdenia 

dominosebri TvalsaCinoeba. klasi kidev rwmundeba, rom raodenoba 

ar icvleba grovis ganlagebis SecvliTa da misi gafarToebiT. 

erTi cvlilebaa: rogorc ki Semogvaqvs ricxvi, momdevno or 

gakveTilze maSinve vswavlobT mis farglebSi Sekrebas, Semdeg 

gamoklebas. esecaa mxolod zepirad, nivTebiT, moqmedebis niSnebisa 

da cifrebis gareSe. ZiriTadi sayrdenia dominosebri TvalsaCinoeba. 

bolos xdeba cxris farglebSi jamebisa da sxvaobebis angariSi 

da pirveli damaxsovreba. cxadia, yvelaferi xdeba cifrul 

CanawerTa gareSe Canawerebi aq mxolod masw.-isTvisaa!

- 122 - 

masw.-ma kidev erTxel unda gadaikiTxos I-2. 

------------------------------------------------------------------------------ 

41/2/! Twjefvmj/! sjdywj! Twjej/! njtj! Tfebsfcb! obtxbwm! sjd. 

ywfcUbo/! yvelaferi keTdeba 12.1-is magvarad. oRond, raki amgvari 

sakiTxi ukve mravaljer damuSavda, sasurvelia daCqareba. 

masw.-i klass dawyvilebiT erTmaneTs adarebinebs momzadebul 

grovebs (ix. I-2). dawyvilebiT aRmoCndeba, rom yvela 

SvideulSi nivTebis raodenoba tolia. _ isrebisa da qoxebis 

raodenoba tolia. isrebi imdenivea, ramdenicaa qoxebi. xuTkuTxedebisa 

da wreebis raodenobac tolia. tolia, magram mainc 

ramdenia? {Svidia}. diax, esec Svideulia da esec. isrebis 

raodenobac Svidia, qoxebisac. xuTkuTxedebi ramdenia? {Svidi}. 


yvela SvideulSi Svidi sagania. Semdeg masw.-i dafasTan 


rgolebis Svideulebi, TiTo moswavlem _ TiTo. Tanac, sxvadasxvanairi 

Svideulebi, sxvisi daxatuli ar gaimeoron. 

mosalod-nelia, rom moswavleebi gaimeoreben im sxvadasxva 


miuTiTebs moswavleTa mier daxatul Svideulebze: _ aqac 

Svidi rgolia, aqac, aqac, aqac. romelia yvelaze lamazi? 

(gamoiTqmeba Sexedulebani) me mgoni, es jobia yvelas (da 

Semoxazavs `dominosebr~ Svideuls). 

Semdeg Svids Sedardeba ukve naswavli ricxvebi: erTi, ori, 

sami, ... eqvsi. amisaTvis dafaze `dominoseburad~ daewyoba 

saTanado grovebi (Svideuli da sameuli da ase Semdeg) da 

dawyvilebiT Sedardeba erTmaneTs. 

_ romeli raodenobaa meti _ Svidi Tu oTxi? {Svidi} ratom? 

{mogvrCa...} romelia naklebi _ Svidi Tu sami? {sami} ratom? ... 

suraTi 30.1. _ ra aris aq? {Svidi patara Savi wre} risi 

Svideulia aq? {wreebis Svideuli} axla Tqven mis gverdiT 

daxazeT lurji wreebis aseTive Svideuli! aq kidev xatia 

sxvadasxva grovebi. jer varskvlavebis grovaa. igi SeaerTeT 

mis Sesabamis raodenobasTan, qvemoT!

- 123 - 

da ase gameordeba blebis, Cxirebisa da burTulebis 

grovebzec. 

_ miCvene Svidi fanqari! daakakune Svidjer! daukari taSi 

Svidjer! 

41/3/! psnbhj! tjwsdjUj! njnbsUfcboj;! xjob! {fnpUb-! vlbob! 

rwfnpUb!eb!njtU/!`!tblvUbsj!Ubwjt!njnbsU/!!!vTqvaT, saklaso 

oTaxSi merxebis ukana kedelze kidia wiwilebis suraTi. 

_ wiwilebis suraTi Tqvens winaa Tu ukan? {ukan} Tqvenze 

ufro zemoTaa Tu qvemoT? {qvemoT} maS wiwilebis suraTi 

yofila ukana zemoTa mxares! Tan ukanaa da Tan _ zemoTa! axla 

miTxariT, ra aris ukana qvemoTa mxares? {iatakze mdebare 

burTi _ masw.-ma winaswar unda dados rame!} es kedeli Tqvengan 

romel mxaresaa? {wina qvemoTa} es carci? {wina zemoTa} ra 

aris kidev Tqvengan wina zemoTa mxares? {sadeni}. 

_ am did kedels (mivuTiTebT oTaxis gverdiTa kedelze!) 

ramdeni kuTxe aqvs? {oTxi} oTarma gvaCvenos yvela! kargi, 

axla TamTam gvaCvenos, romelia wina qvemoTa kuTxe! ukana 

zemoTa? gigi, romelia am kedlis ukana qvemoTa kuTxe? wina 

zemoTa kuTxe? 

_ xvalisaTvis moigoneT amocana wina-ukanasa da zemoTaqvemoTas 

Sesaxeb! 

41/4/!TwjelvUyfej!)Db{ofrjmj*/!njtj!ebyb{wb!)njbympfcjU*/! 

suraTi 30.3. _ ramdenkuTxedia es wiTeli nakvTi? 

daTvaleT misi gverdebi! {SvidkuTxedia}. mis gverdiT 

mkrTalad moniSnulia xazebi da wertilebi. erTze daxazeT 

SvidkuTxedi! sxvebi _ Sin! gverdiT rom sicarielea, Sin kidev 

erTi axali SvidkuTxedi daxazeT. saxazaviT! maS, sul ramdeni 

SvidkuTxedi iqneba am suraTze? {eqvsi}. 

41/5/ `txsbgj!qbtvyfcj!`!lwjsjt!eSffcj~/ TamaSamde vambobT: 

_ vin icis, ramdeni dRea kviraSi? {Svidi dRea}. davTvaloT: 

orSabaTi, samSabaTi, oTxSabaTi, xuTSabaTi, paraskevi, SabaTi, 

kvira (bavSvebi Tan imeoreben dReebis saxelwodebebs, Tan 

TiTebze iTvlian). 

Semdeg TamaSi: rig-rigobiT unda daasaxelon kviris dReebi.

- 124 - 

pirveli moswavle ambobs òrSabaTi~, meore _ `samSabaTi~, da 

ase Semdeg, merve moswavle isev _ òrSabaTi~ da ase Semdeg 

wriulad. tempi TandaTan Cqardeba. 

g 31.!Tflsfcb/!psnbhj!tjwsdjUj!njnbsU/!tywbhwbsj!TwjelvUyfe/! 


42/2/!Tflsfcb: 6+1, 1+6, 5+2, 2+5, 4+3, 3+4. masw.-is daxmarebiT 

bavSvebi Slian mTw-s `7~-ze. _ aq ramdeni wrea? {Svidi} aq 

ki ramdeni wrea? {aqac Svidi} Svidi wre rogor aris aq 

dawyobili? {oTxi da sami}. axla davTvaloT wreebis raodenoba 

aq! ramdeni wrea? {Svidi} aq rogor aris dawyobili 

Svidi wre? {xuTi da ori} aq ramdeni wrea? {aqac Svidi} aq 

rogorRa aris dawyobili Svidi wre? {eqvsi da erTi}. 

moswavleebs wigns davaxurinebT da mexsierebiT, 

warmodgeniT vapasuxebinebT: _ rogor iyo dalagebuli Svidi 

wre? {oTxi da sami} kidev rogor iyo? {xuTi da ori} kidev? 

{eqvsi da erTi}. 


dafaze `dominoseburad~ ideba eqvseuli da erTeuli. 

gamoZaxebuli moswavle krebs magnitebs. _ eqvsisa da erTis 

SekrebiT miviReT Svidi. eqvs mivumatoT erTi udris... {Svids}. 

igive magnitebi piriqiT gadaewyoba _ erTeuli da eqvseuli. 

_ risi toli iqneba erTs mivumatoT eqvsi? {Svidisa} saidan 

ici? {erTs mivumatoT eqvsi igivea, rac eqvss mivumatoT erTi, 

sulerTia}. 

aseve Seikribeba 5+2, 2+5, 4+3 da 3+4. 

mTw isev gadaiSleba `7~-ze. _ aq rogoraa dawyobili 

Svidi wre? {eqvsi da erTi} ramdeni miiReba eqvsisa da erTis 

SekrebiT? {Svidi} erTisa da eqvsis SekrebiT? {igive Svidi} 

kidev risi da risi SekrebiT miviRebT Svids? aba naxeT 

sxvagan, rogoraa dawyobili Svidi wre! {oTxi da sami; xuTi 

da ori}. risi toli yofila ors mivumatoT xuTi? sams 

mivumatoT oTxi? 

Tan pasiur bavSvebs wreebi unda davaTvlevinoT àqedan

- 125 - 

iqiT~ (marcxnidan marjvniv) da ùkuRma~ (marjvnidan marcxniv). 

albaT gaaqtiurdebian. sxvebma amasobaSi SekiTxvebs upasuxon. 

42/3/!psnbhj!tjwsdjUj!njnbsUfcboj;!{feb!nbskwfob-!xjob!nbsd. 

yfob!eb!njtU/!`!tblvUbsj!Ubwjt!njnbsU/!_ Savi skami Sens winaa 

Tu ukan? {ukan} Sens marjvnivaa Tu marcxniv? {marcxniv} maS 

Savi skami yofila ukana marcxena mxares! Tan ukanaa da Tan _ 

marcxena! axla miTxariT, ra aris ukana marjvena mxares? 

{iatakze mdebare burTi _ masw.-ma winaswar unda dados rame!} 

es kuTxe Tqvengan romel mxaresaa? {wina marcxena} aris kidev 

Tqvengan wina marcxena mxares? {sinaTlis CamrTveli}. giorgi 

Sens romel mxares zis? {wina marcxena} vin zis Sens wina 

marjvena mxares? (da ase Semdeg). 

_ am dafas ramdeni kuTxe aqvs? {oTxi} oTarma gvaCvenos 

yvela! kargi, axla TamTam gvaCvenos, romelia qvemoTa marjvena 

kuTxe! zemoTa marjvena? gigi, romelia am dafis qvemoTa 

marjvena kuTxe? zemoTa marcxena kuTxe? 

_ sadaa am oTaxis Weri? Wers ramdeni kuTxe aqvs? {oTxi} 

gvaCvene yvela! kargi, axla gvaCvene, romelia wina marjvena 

kuTxe! (da ase Semdeg). 

bolos, cariel CarCoSi 31.2 maswavlebis karnaxiT Caxataven: 

_ CarCos SuagulSi CaxateT wiTeli wre, arc Zalian didi da 

arc Zalian patara. moZebneT CarCos zeda marjvena kuTxe! iq 

daxateT patara mwvane samkuTxedi. moZebneT CarCos zeda 

marcxena kuTxe! iq daxateT patara yviTeli samkuTxedi. 

moZebneT CarCos qveda marjvena kuTxe! iq daxateT patara 

lurji samkuTxedi. moZebneT CarCos qveda marcxena kuTxe! iq 

daxateT patara Savi samkuTxedi. axla mwvane samkuTxedis 

marcxniv daxateT mwvane kvadrati... (da ase Semdeg _ masw.-i 

karnaxobs nakvTebs ise, rom mTeli CarCo gaivsos da bolos 

lamazi Wreli suraTi miviRoT). 

42/4/! tywbhwbsj! TwjelvUyfefcj/! nbUj! ebyb{wb! )njbympfcjU*/!! 

masw.-i dafaze erT striqonSi daxazavs Svid nakvTs: arasimetriul 

oTxkuTxeds, kvadrats, arasimetriul xuTkuTxeds, arasimetriul 

eqvskuTxeds, erT simetriul Cazneqil da erT arasi-

- 126 - 

metriul amozneqil SvidkuTxeds (nimuSebi ix. suraTze 31.3). 

_ ramdeni nakvTia dafaze? {Svidi} pirveli nakvTi ra 

nakvTia? {oTxkuTxedi} ratom? {oTxi gverdi aqvs} meore? 

{kvadratia, Tan oTxkuTxedicaa} diax, Cven xom viciT, rom 

kvadrati Tan oTxkuTxedicaa! mesame ra nakvTia? {xuTkuTxedi} 

saidan iciT? {xuTi gverdi aqvs} diax, raki xuTi gverdi aqvs, 

amitom xuTkuTxedia. meoTxe ra nakvTia? {eqvskuTxedi} saidan 

iciT? {eqvsi gverdi aqvs} diax, raki eqvsi gverdi aqvs, amitom 

eqvskuTxedia. meeqvse ra nakvTia? {SvidkuTxedi} saidan iciT? 

{Svidi gverdi aqvs} diax, raki Svidi gverdi aqvs, amitom 

SvidkuTxedia. es ukanaskneli, meSvide, ra nakvTia? 

{SvidkuTxedi} saidan iciT? {Svidi gverdi aqvs} diax, raki 

Svidi gverdi aqvs, amitom esec SvidkuTxedia. ramdeni 

oTxkuTxedi yofila dafaze? {ori} xuTkuTxedi? {erTi} 

eqvskuTxedi? {erTi} SvidkuTxedi {ori}. 

_ axla daakvirdiT suraTs (31.3). ramdeni wiTeli nakvTia? 

{ori} pirveli ra nakvTia? {SvidkuTxedi} ratom? {Svidi 

gverdi aqvs} diax, raki Svidi gverdi aqvs, amitom 

SvidkuTxedia. meore ra nakvTia? {SvidkuTxedi} ratom? {amasac 

Svidi gverdi aqvs}. 

_ axla Tqven wertilebze gadaaxazeT da gaaferadeT 

sxvadasxvagvari SvidkuTxedi. mere gaferadeba ar dagaviwydeT! 

Sin, cariel adgilas daxazon TavianTi mogonebuli axali 

SvidkuTxedi, sxvebic daxazon (31.6). 

42/5/! `txsbgj! qbtvyfcj! `! dysjmj~/ msr-Si vnaxavT naswavl 

cxrils 29.3. SekiTxvebi ki iseTivea, rogorebic iyo 28.3-Si. 

pirvel wreze: _ ra ferisaa mesame striqoni? meramdenea ruxi 

striqoni? risi svetia meeqvse? meramdenea Tevzebis sveti? ... 

momdevno wreebze SekiTxvebi rTuldeba: _ ra xatia mesame 

striqonis meoTxe ujredSi? mesame striqonis meoTxe ujredi! 

{wiTeli kubi}. saWiroa iTqvas ferica da saganic. garda amisa, 

emateba SekiTxvebi cxrilis zeda marjvena, zeda marcxena, 

qveda marjvena da qveda marcxena ujredebis Sesaxeb. 

g 32.!hbnplmfcb/!tjwsd/!tywjt!njnbsU/!TwjelvUyfefcjt!ebzpgb/!

- 127 - 


43/2/!hbnplmfcb;!8`2-!8`7-!8`3-!8`6-!8`4-!8`5/! jer swrafad 

gameordeba 31.1-is Sekrebis magaliTebi, sasurvelia, 

zepirad. visac gauWirdeba, gadaSalos mTw. 

dafaze `dominoseburad~ ideba Svideuli. gamoZaxebuli 

moswavle mas gamoaklebs erTs. _ ramdeni miviReT? {Svids 

gamovakloT erTi udris... eqvss}. aseve Sesruldeba danarCeni 

gamoklebanic. 

mTw gadaiSleba `7~-ze. _ aq rogoraa dawyobili Svidi 

wre? {eqvsi da erTi} ramdeni miiReba, Svids rom erTi 

gamovakloT? aba, erT wres daafareT TiTi da daTvaleT! 

{eqvsi} ramdenia Svids gamovakloT eqvsi? {erTi} Svids 

gamovakloT ori? Svids gamovakloT xuTi? Svids gamovakloT 

sami? Svids gamovakloT oTxi? 

yoveli pasuxi unda aixsnas: Svidi wre dawyobilia eqvsad da 

erTad. Tuki eqvs wres movacilebT, darCeba erTi; Tuki erT wres 

movacilebT, darCeba eqvsi. yvela magaliTi aseve ganixileba. 

43/3/!psnbhj!tjwsdjUj!njnbsUfcboj!tywjt!njnbsU/!keTdeba igive, 

rac 30.2 da 31.2-Si, oRond moswavlem unda warmoidginos 

es mimarTebani sxva moswavlis TvalTaxedviT. magaliTad: _ 

Weris romeli kuTxea wina marjvena giorgisaTvis? am kedlis 

romeli kuTxea ukana zeda SorenasaTvis? 

am gakveTilze marjvena/marcxena mimarTebis momcveli 

SekiTxvebi daismis mxolod im SemTxvevaSi, roca ori moswavle 

erTmaneTis mimarT zurgiT arian _ asecaa bunebrivad merxebze 

jdomisas. xolo roca meore adamiani bavSvis mimarT 

pirisaxiTaa _ misi marjvena/marcxena piriqiTaa! es Zneli 

SemTxvevaa da SemdgomSi iqneba. 

43/4/!TwjelvUyfefcjt!ebzpgb!tywb!nsbwbmlvUyfefcbe/! 

_ daakvirdiT Tqvens mier gaferadebul nakvTebs (suraTi 

32.3). ra nakvTebia? {SvidkuTxedebi} ramdenia? {xuTi} jer 

vnaxoT pirveli SvidkuTxedi. igi gayofilia or nakvTad. 

radara nakvTebadaa igi gayofili? daTvaleT maTi gverdebis

- 128 - 

raodenoba! {oTxkuTxedi da xuTkuTxedi} axla vnaxoT meore 

SvidkuTxedi. radara nakvTebadaa igi gayofili? daTvaleT maTi 

gverdebis raodenoba! {samkuTxedi da SvidkuTxedi} axla mesame 

SvidkuTxedi gavyoT samkuTxedad da eqvskuTxedad (Tan 

daixazeba dafaze). rogor gavyoT? 

amisaTvis Sua qimis erTerTi gverdi unda gavagrZeloT. 

_ axla vnaxoT meoTxe SvidkuTxedi. radara nakvTebisaganaa 

igi Sedgenili? daTvaleT maTi gverdebis raodenoba! {oTxkuTxedisagan 

da xuTkuTxedisagan} axla vnaxoT mexuTe SvidkuTxedi. 

radara nakvTebisaganaa igi Sedgenili? daTvaleT maTi 

gverdebis raodenoba! {ori cali xuTkuTxedisagan} axla gverdiT 

cariel adgilas Tqven TviTon daxazeT axali SvidkuTxedi 

da gayaviT igi or nakvTad. naxeT, radara nakvTebs miiRebT. 

43/5/! `txsbgj! qbtvyfcj~!` ramdeni TiTia. ix. 22.4. oRond, 

TiTebis raodenoba aRwevs Svidamde. 

43/9/ Tuki moswavleebs gauWir- 

debaT, saWiroa moZravi nakvTebis 

mixmareba. moswavleebma magnitur 

dafaze scadon kvadratisTvis 

samkuTxedis midgma da Seadaron erTmaneTs oTxi SesaZlebloba 

_ gaarkvion, ramdenkuTxedi miiReba TviTeul SemTxvevaSi. 

g 33.!swbfvmj-!swb/!nbsk//nbsdy/.jt!!Tfcsvofcb/!swblvUyfej/! 

44/2/! swbfvmj/! sjdywj! swb/! njtj! Tfebsfcb! obtxbwm! 

sjdywfcUbo/! yvelaferi keTdeba iseve, rogorc 30.1-Si, iseve 

SedarebiT aCqarebulad. 

emateba SekiTxva: _ Tu iciT, romel cxovelebs aqvT rva 

fexi? {rvafexas, zRvaSi rom cxovrobs; obobas; kidev _ 

kiborCxalas!}. 

44/3/!nbskwojw!/!nbsdyojw!njnbsUfcjt!Tfcsvofcjt!TfnUywfwfcj! 

)qjsjtqjs! ehpnjtbt*/! jer gavimeorebT marjvena/marcxena mimar- 

Tebas sxvis mimarT, roca es sxva zurgSeqceviTaa da misi marjvena/marcxena 

emTxveva Cvensas. xolo roca meore adamiani 

bavSvis mimarT pirisaxiTaa _ misi marjvena/marcxena piriqiTaa!

- 129 - 

es Zneli SemTxvevaa da sagangebo swavleba esaWiroeba. 

gamoviyvanT or moswavles da davayenebT erTmaneTis 

pirispir. _ romelia Sengan marcxena mxare? Sengan? 

(aRmoCndeba, rom sxvadasxvaa) ratom moxda, rom marcxena mxare 

beqasTvis aqeTaa, xolo bakurisTvis _ iqiT? 

es sityvebiT Zneli asaxsnelia, magram mTavaria, rom yvelam 

kargad gaigos, rac xdeba. aseve _ marjvena mxarisTvisac. _ 

roca ori adamiani saxiT erTmaneTis pirispiraa, maTi marjvena 

da marcxena mxareebi erTmaneTis sapirispiroa. es kargad 

daimaxsovreT da ar SegeSaloT xolme! magram roca adamianebi 

zurgiT arian? {maSin maTi marjvena mxareebi igivea, marcxena 

mxareebic igivea} axla miTxariT, ra aris Cemgan marjvniv? (am 

dros masw.-i dgas pirisaxiT klasisaken) ra aris Cems marcxniv? 

{larnaki}. axla daakvirdiT suraTs (33.2). gavarkvioT, kedeli 

biWunasgan romel mxaresaa? Cveni mxareebi ar mikiTxavs, me 

biWunas marjvena/marcxena gkiTxeT! {kedelia biWunasgan 

marcxniv} biWunasgan marjvniv ra aris? {skami}. 

Semdeg iqve cariel CarCoSi masw.-is sityvieri karnaxiT (da 

ara CvenebiT) Caxataven: SuagulSi kacunas; kacunasagan marjvniv 

_ samkuTxedebisagan Sedgenil naZvs; kacunasagan marcxniv _ 

kvadratebisgan Sedgenil kedels; zemoT... zemoTa marjvena 

kuTxeSi... qvemoTa marcxena kuTxeSi... da ase Semdeg _ masw.-is 

mier mogonebul geometriul naxatebs. 

44/4/!swblvUyfej/!njtj!ebyb{wb!)njbympfcjU*/! 

yvelaferi keTdeba iseve, rogorc 30.3-Si.! 

44/5/!`txsbgj!qbtvyfcj!`!dysjmj~. msr-Si vnaxavT naswavl 

cxrils 29.7. SekiTxvebi ki iseTivea, rogorebic iyo 31.4-Si. 

44/8/!Tuki gauWirdebaT, dafaze daixazos didi eqvskuTxedi 

da moswavlem scados misi sxvadasxvagvari gayofa. gamodgeba 

gayofa Sua diagonaliT (es termini moswavles ar sWirdeba!).!! 

44/8/!yuradReba miaqcion naxazze mkrTal xazs! 

g 34/!Tflsfcb/!hbnfpsfcb/!wbstlwmbwjtfcvsj!swblvUyfefcj/ 

45/2/!Tflsfcb: 7+1, 1+7, 6+2, 2+6, 5+3, 3+5, 4+4.!yvelaferi keT-

- 130 - 

deba iseve, rogorc 31.1-Si, SedarebiT aCqarebulad. 

SekiTxvaTa nimuSebi mas Semdeg, rac mTw gadaiSleba `8~-ze: 

_ ramdeni miiReba Svidisa da erTis, erTisa da Svidis 

SekrebiT? SevamowmoT! ramdeni miiReba eqvisi da oris 

SekrebiT? orisa da eqvsis SekrebiT? SevamowmoT! ramdeni 

miiReba xuTisa da samis SekrebiT? samisa da xuTis SekrebiT? 

SevamowmoT! risi tolia oTxs mivumatoT oTxi? ... 

45/3/!Ubwj!5/3.jt!hbnfpsfcb!eb!Tfkbnfcb/! 

jer swrafad gavimeorebT 41.2-s: _ cisana, vin aris Cemgan 

marjvniv? guram, axla vin aris Cemgan marcxniv? (da ase 

ramdenjerme _ masw.-i klasisken pirisaxiTaa). 

_ axla davaTvalieroT suraTi (42.2). ras xedavT masze? 

{gogona quCis piras dgas} diax, es gogona skolidan Sin brundeba. 

misi saxli gorazea. axla me getyviT, rogor unda iaros 

gogonam, Tqven fanqris wveri gaayoleT gzas, yuradRebiT, ar 

SegeSaloT! vnaxoT, romelia gogonas saxli! gza Zalian ar daxazoT, 

cotaTi, mkrTalad gaayoleT fanqari, imitom, rom me 

mere sxva gzasac gikarnaxebT! yuradReba miaqcieT marto gzas, 

sxva ramem ar dagabnioT! yvela gza vardisferia! 

masw.-i karnaxobs: _ gogona gadavida quCaze, gauxvia 

marcxniv, mere Seuxvia quCaze, gahyva quCas, gaiara erTi 

Sesaxvevi, ar Seuxvia, meore, mesame Sesaxvevi... mere iq marjvniv 

gauxvia ... (da ase Semdeg _ masw.-ma Tavad SearCios svlagezi, 

romelic mkafiod aRiwereba). ai, miadga Tavis saxls da Sig 

Sevida! aba, SemoxazeT wiTlad gogonas saxli! vnaxoT, vis 

SeeSala da vis _ ara. 

_ axla daviwyoT Tavidan. gogona dRes midis Tavis 

bebosTan da papasTan. me gikarnaxebT gzas, Tqven gaayoleT 

fanqari da moZebneT bebosa da papas saxli! (movaZebninebT da 

mwvaned SemovaxazvinebT). 

aseve moZebnian mamidis saxls da lurjad Semoxazaven. 

samive SemTxvevaSi masw.-i aarCevs zemoTa, SedarebiT Znelad 

misagneb saxlebs. gzadagza ki amgvar SekiTxvebs svams: _ axla 

ra aris gogonas win? gogonas ukan? marcxniv? marjvniv?

- 131 - 

45/4/!tywbhwbsj-!wbstlwmbwjtfcvsj!swblvUyfefcj/!nbUj!ebyb{. 

wb!)njbympfcjU*/!!yvelaferi keTdeba iseve, rogorc 31.3-Si. 

45/5/ isev suraTi 34.2. masw.-i miuTiTebs erTerT sxva, 

Semouxazav saxlze. _ ai es ki gogonas klaseli megobris 

saxlia. SemoxazeT yviTlad! gogona dRes megobarTan unda 

mivides. axla Tqven rig-rigobiT unda uTxraT gogonas, rogor 

iaros. visac SeeSleba, momdevno moswavle Seusworebs. 

daiwyebs naTela! {gadadi quCaze} aleqsandre, axla Sen 

ukarnaxe erTi rame! {gauxvie marcxniv} givi, kargad ukarnaxa 

aleqsandrem? {ara, marjvniv unda gadaexvia}... (da ase Semdeg) 

Tuki moeswreba, kidev erTi saxls `miagnebineben~ gogonas. 

g 35. hbnplmfcb/!.jU!nfu.oblm/!ebxzwjmfcjU/!swblvU/!ebzpgb/ 

46/1/!45/8 _!SemoqmedebiTia! 

45/9-is davaleba kompleqsuria. awi TiTqmis yovel gakve- 

Tilze iqneba amgvari davalebani, romlebic TandaTan garTuldeba 

kidec _ maT Sesaxeb ix. § 5-is dasawyisSi. magaliTad, 

34.8-s SemTxvevaSi masw.-i gamokveTs: _ ramdeni rame gvevaleboda 

am amocanaSi? radara gvevaleboda? jer ra unda SevasruloT? 

{amovarCioT SvidkuTxedebi da gavaferadoT isini} diax, da am 

dros ar unda vifiqroT arc rvakuTxedebze da arc 

dawyvilebaze! SvidkuTxedebis gaferadebas rom movrCebiT, 

Semdeg ra unda SevasruloT? {amovarCioT rvakuTxedebi da gavaferadoT 

isini} diax, da am dros unda daviviwyoT SvidkuTxedebi. 

Semdeg ra unda gavakeToT? {davawyviloT SvidkuTxedebi 

da rvakuTxedebi} da am dros unda daviviwyoT gaferadeba! 

bolos ki ra unda gavakeToT? {davakvirdeT, romelia meti} 

maS, sul ramdeni rame gvevaleboda am amocanaSi? CamoTvaleT! 

Sesrulebisas ramdeni nabiji iyo? {oTxi} CamovTvaloT mokled 

(Tan aCvenebs TiTebs da TiTebiT iTvlis): pirveli _ SvidkuTxedebis 

gaferadeba; meore _ rvakuTxedebis gaferadeba; 

mesame _ dawyvileba; meoTxe _ metobis garkveva. 

aseve unda damuSavdes momdevno gakveTilebis kompleqsuri 

amocanebic, Tuki maTi amoxsna moswavleebs gauWirdebaT.

- 132 - 

46/2/!hbnplmfcb;!9`2-!9`8-!9`3-!9`7-!9`4-!9`6-!9`5/!yvelaferi 

keTdeba iseve, rogorc 32.1-Si, SedarebiT aCqarebulad, jer 

34.1-is swrafi zepiri gameorebiT. 

SekiTxvaTa nimuSebi mas Semdeg, rac mTw gadaiSleba `8~-ze: 

_ ramdeni wrea aq? {rva} rogoraa rva wre dawyobili aq? 

{Svidi da erTi} aq? ... ramdenia rvas gamovakloT erTi? rvas 

gamovakloT Svidi? rogor mixvdi? {rva dawyobilia Svidad da 

erTad. rva wridan aviRebT Svids, darCeba erTi...}. 

46/3/!`.jU!nfu.oblmfcpcb~!xbsnpehfojU/!`.jU!nfupcb~!eb!`. 

jU!oblmfcpcb~!ebxzwjmfcjt!Tfefhbe/! 

aq erTi siaxlea: dawyvilebiT varkvevT ara mxolod imas, 

Tu romelia meti, aramed agreTve imasac, Tu ramdeniTaa meti. 

jer TiTo-TiTo moswavles swrafad gavameorebinebT `-iT 

met-naklebobas~ sxvadasxva doneebze: 1) nivTTa grovaze `-iT 

met-naklebi~ nivTebis dawyoba; 2) dafaze nakvTebis grovaze ìT 

met-naklebi~ nakvTebis daxatva; 3) masw.-is mier dafaze 

daxatul `dominoze~ `-iT met-naklebi~ `dominos~ daxatva. 

masw.-i Tan ekiTxeba: _ saidan ici, rom marTlac 4-iT metia? 

{davxate imdenive da Semdeg kidev davumate oTxi} saidan ici, 

rom marTlac 5-iT naklebia? {davxate imdenive da Semdeg 

movacile xuTi}. 

_ axla warmoidgineT, rom klasis gogonebi dawyvildnen 

biWebTan da TamaSi daiwyes. magram ori gogona mewyvile biWis 

gareSe darCa. klasSi gogonebi yofilan meti Tu biWebi? ramdeniT? 

saidan ici? 

amasobaSi masw.-i dafaze daxatavs daaxloebiT 18 rgolsa da 

17 marTkuTxeds. daavalebs moswavles, dawyvilebiT gaarkvios, 

romelia meti da ramdeniT. moswavle daawyvilebs (xazebiT 

SeerTebiT) da aRmoCndeba, rom uwyvilod morCa erTi rgoli. 

_ romeli yofila meti? {rgolebi} ramdeniT meti yofila? 

{erTiT} saidan ici? 

_ axla Zalian swrafad unda mipasuxoT. vnaxoT, xels vin 

aiwevs pirvelad! msr-Si suraTze (35.2) romelia meti da 

ramdeniT _ wreebi Tu samkuTxedebi? saidan ici?

- 133 - 

Semdeg masw.-i moyveba amocanas: _ magidaze eqvsi kalmistari 

ido. movida oTxi moswavle da TiTom TiTo kalmistari aiRo. 

darCa Tu ara magidaze kalmistrebi? romeli iyo meti _ 

kalmistrebi Tu moswavleebi? 

rogorc yvela sayofacxovrebo amocanas, amis amoxsnasac 

pirobis gameorebiTa da gaazrebiT viwyebT. mTavaria, klasma 

kargad gaiazros is, Tu ras niSnavs `TiTom TiTo~. 

Tuki gauWirdebaT pirobis gaazreba an amocanis amoxsna 

zepirad, dafaze daixateba sqema: viTom monakveTebi _ 

kalmistrebia, rgolebi _ moswavleebi da maT xazebiT 

SeaerTeben dasawyvileblad. 

46/4/!swblvUyfefcjt!ebzpgb!tywb!nsbwbmlvUyfefcbe/! 

yvelaferi keTdeba iseve, rogorc 32.3-Si. 

46/5/!`txsbgj!qbtvyfcj~!`!ÙjUfcjt!Dwfofcb~/ ix. 20.4. masw.-i 

moswavles usaxelebs ricxvs (2, 3, 4, 5, 6, 7 an 8). moswavlem 

unda aCvenos amdeni TiTi, oRond aucileblad orive xelis 

TiTebi _ ise, rom erTad iyos dasaxelebuli ricxvi. 

46/9/! Tuki gauWirdebaT, dafaze daixazos didad da 

moswavlem scados misi sxvadasxvagvari gayofa. gamodgeba 

gayofa. _ ras miviRebT, mTlianad rom movaWraT zeda 

oTxkuTxedi? {or oTxkuTxeds} Cven ki ra gvWirdeba? maS ra 

movaWraT? {sul zemoT patara oTxkuTxedi}. 

g 36.!dysbfvmj-!dysb/!nfu.oblm/!tbz/!bnpdbofcTj/!dysblvUyfej/! 

47/1/!46/9 _ SemoqmedebiTia. 35.7 _ Zalian sainteresoa. 

46/7-is davaleba kompleqsuria. Tuki misi amoxsna 

moswavleebs gauWirdaT, igi unda damuSavdes iseve, rogorc 

34.8. aseve unda damuSavdes momdevno gakveTilebis kompleqsuri 

amocanebic, Tuki maTi amoxsna moswavleebs gauWirdebaT. 

47/2/! dysbfvmj/! sjdywj! dysb/! njtj! Tfebsfcb! obtxbwm! sjd. 

ywfcUbo/!!!yvelaferi keTdeba iseve, rogorc 30.1-Si. 

47/3/!nfu.oblmfcpcb!vnbsujwft!tbzpgbdypwsfcp!bnpdbofcTj/! 

gavameorebinebT `-iT met-naklebobis~ garkvevas dawvilebis

- 134 - 

Sedegad (35.2): jer erTxel dafaze, Semdeg ki msr-Si suraTze 

(36.2). gairkveva, wreebia meti Tu marTkuTxedebi da ramdeniT. 

Semdeg amovxsniT kidev erT imdagvar amocanas, rogoric 

iyo 35.2-is bolos: 

_ ciras dabadebis dReze dedam xuTi xaWapuri gamoucxo. 

ciras stumrad Svidi megobari ewvia. eyofaT Tu ara bavSvebs 

xaWapurebi, TiTom rom TiTo xaWapuri SeWamos? 

pirobis gaazrebisa da amocanis amoxsnis Semdeg daemateba 

SekiTxva: _ aba ra qnan bavSvebma, sxva saWmelebi ki aris, 

magram xaWapuri rom yvelas unda? {gaiyon} marTalia, maSin Ti- 

Tos TiTo xaWapuri ar Sexvdeba, unda gaiyon. Tu eyofaT bavSvebs 

xaWapurebi, TiTo xaWapuri rom or-orma bavSvma gaiyos? 

sakmaod Zneli SekiTxvaa, amitom saWiro iqneba TvalsaCino 

sqema: dafaze daixateba xuTi rgoli (viTom xaWapurebia) da 

zemodan patara xazebi (viTom bavSvebia). _ ramdeni xazi unda 

davxazoT? {mosalodnelia pasuxi `Svidi~, magram masw.-i 

Seaxsenebs, rom TviTon cirasac surs xaWapuri!} sul ramdeni 

bavSvia? {rva} axla es xaWapurebi viTom Suaze gaiyves 

(monakveTebiT rgolebs dayofs). davawyviloT da vnaxoT, Tu 

eyofaT (xazebiT SeaerTeben). 

_ moswavlem diliT ramdenime naxati daxata, saRamos kidev 

sami naxati daxata. rogorc Seicvala naxatebis raodenoba 

saRamos? moimata Tu moiklo? {moimata} ramdeniT? {samiT}. 

axla Tqven moigoneT iseTi ambavi, rom raimes raodenoba oTxiT 

imatebdes. oRond, arc moswavleze iyos da arc daxatvaze! 

masw.-i ramdenime moswavles mousmens, Seecdeba, rom 

viTarebani sul sxvadasxva sayofacxovrebo Sinaarsisa iyos. 

Semdeg òTxiT metobas~ Secvlis èqvsiT metobiT~, `samiT 

metobiT~ da ase Semdeg. 

Semdeg gakeTdeba igive, oRond raodenobis moklebaze. masw.iseuli 

nimuSi: _ mirians mravali saTamaSo hqonda da amitom 

oTxi saTamaSo biZaSvils aCuqa. moiklo Tu moimata mirianis 

saTamaSoebis raodenobam? ramdeniT? 

47/4/!dysblvUyfej/!njtj!ebyb{wb!)njbympfcjU*/!

- 135 - 


47/5/! `nbubsfcmpcbob~/ igive TamaSi, rac 14.4, oRond 

vagonTa raodenoba ivseba cxramde (eqvsis nacvlad). 

g 37. Tflsfcb/!tbzpgbdy/!bnpdbo/!tywbhwbsj!dysblvUyfefcj/! 

48/2/!Tflsfcb: 8+1, 1+8, 7+2, 2+7, 6+3, 3+6, 5+4, 4+5.!yvelaferi 

keTdeba iseve, rogorc 34.1-Si, SedarebiT aCqarebulad. 

SekiTxvaTa nimuSebi mas Semdeg, rac mTw gadaiSleba `9~ze: 

_ aq ramdeni wrea? {cxra} cxra wre rogoraa dawyobili 

aq? {rva da erTi} ramdenia rvas mivumatoT erTi? erTs 

mivumatoT rva? ... Svids mivumatoT ori? ors mivumatoT 

Svidi? eqvss mivumatoT sami? sams mivumatoT eqvsi? xuTs 

mivumatoT oTxi? oTxs mivumatoT xuTi? rogor mixvdi, 

ramdenia xuTs mivumatoT oTxi? (moswavlem unda uCvenos masw.s 

xuTeulad da oTxeulad dawyobili cxraeuli). 

48/3/! sbpefopcjt! nbufcb! vnbsujwft! tbzpgbdypwsfcp! bnp. 

dbofcTj/!jer amoixsneba imdagvarive amocana, rogoric iyo 36.2- 

Si (ciras dabadebis dReze), oRond xaWapurebi Seicvleba xeze 

msxlebiT: xeze mxolod oTxi didi msxali darCa, bavSvebi ki 

Svidni mogrovdnen... 

amocanebi, rogorc yovelTvis, safexurebad amoixsneba. 

Semdgom amocanebs winaswar emateba SekiTxva imis Sesaxeb, 

raodenoba gadidda Tu Semcirda, da ramdeniT. amasTan, 

meorejer dasaxelebuli ricxvi naklebia pirvelze! 

1) dodom erT dRes Tojinas xuTi kaba Seukera, meore 

dRes ki _ erTi. Tojinis kabebi meore dRes meti gaxda Tu 

naklebi? {meti gaxda} moimata Tu moiklo dodos Tojinis 

kabebis raodenobam? {moimata, imitom rom meore dRes dodom 

kidev erTi kaba daumata, pirvel dRes Sekerilebic xom 

Senaxuli hqonda!} dodos Tojinas pirvel dRes ufro meti 

kaba eqneboda Tu meore dRes? {meore dRes} ramdeniT? {erTiT} 

sul ramdeni kaba Seukeravs dodos? {eqvsi} rogor gaige? 

{xuTs mivumate erTi}. 

2) masw.-ma dafaze sami wre daxaza, Semdeg moswavlem _ ori

- 136 - 

kvadrati. moimata Tu moiklo dafaze nakvTebis raodenobam? 

{moimata} ramdeniT? {oriT} sul ramdeni nakvTi daxazula 

dafaze? {xuTi} rogor gaige?... 

3) dedalma eqvsi kvercxi dado sabudarSi da gamosaCekad 

daajda, sami kvercxic sxva dedlisa Caudes. moimata Tu 

moiklo kvercxebis raodenobam? {moimata} ramdeniT? {samiT} 

sul ramden kvercxs azis mokruxebuli dedali? {cxras} 

rogor gaige?... 

Tan dafaze sqemebi ixazeba. magaliTad, daixateba xuTi 

samkuTxedi (ès viTom kabebia~) da Semdeg calke kidev _ erTi 

samkuTxedi. aseve, wreebi da kvadratebi, aseve, elifsebi 

(romlebic viTom kvercxebia). 

Semdeg masw.-i mohyveba sxvagvar amocanebs. 

4) gias da daTos misces erTidaimave raodenobis 

CurCxelebi. mere gias kidev misces sami CurCxela. romels aqvs 

axla meti CurCxela? 

masw.-i svams kiTxvebs: _ Tavidan romels hqonda meti, gias 

Tu daTos? {Tavidan gias hqonda imdenive CurCxela, ramdenic 

daTos}. vis daumates CurCxelebi, gias Tu daTos? ramdeni 

daumates? ras gvekiTxebian? {sabolood romels aqvs meti, 

gias Tu daTos}. rogoria amocanis pasuxi? (gias aqvs meti 

CurCxela, vidre daTos). ramdeniT meti? {samiT}. saidan ici? 

{jer tolad hqondaT, Semdeg gias kidev sami daumates}. 

5) klasSi gogonebi da biWebi toli raodenobiT iyvnen, ese 

igi, gogonebi imdenive iyvnen, ramdenic _ biWebi. Semdeg klasSi 

daemateT ori gogona. romeli gaxda naklebi? ramdeniT? (aq 

iyos swored `naklebi~, nacvlad `metisa~). 

esec winas magvarad amoixsneba. 

48/4/!tywbhwbsj!dysblvUyfefcj/!nbUj!ebyb{wb!)njbympfcjU*/! 


48/5/! `txsbgj! qbtvyfcj~! `! nbskwfob! /! nbsdyfob. _ miCvene 

marjvena fexi! marcxena yuri! marjvena mxari! Seni mezobeli 

marcxniv! (da ase Semdeg). tempi TandaTan matulobs. 

48/9/!` aq daxatulia daxveuli xazi. xedavT mis Tavsa da

- 137 - 

bolos? Sin, roca am amocanis amoxsnas daiwyebT, ar 

dagaviwydeT xazis Tavi da bolo! 

g 38. hbnplmfcb/!tbzpgbdy/!bnpdbo/!dysblvUyfejt!ebzpgb/ 


49/2/! hbnplmfcb;! :`2-! :`9-! :`3-! :`8-! :`4-! :`7-! :`5-! :`6/! 

yvelaferi keTdeba iseve, rogorc 35.1-Si, SedarebiT 

aCqarebulad. 

SekiTxvaTa nimuSebi mas Semdeg, rac mTw gadaiSleba `9~ze: 

_ rogor aris aq dawyobili cxra? {rva da erTi} ramdenia 

cxras gamovakloT erTi? cxras gamovakloT rva? rogor 

mixvdi? {cxra wre dawyobilia rvad da erTad, cxridan rvas 

rom aviRebT, darCeba erTi. cxridan erTs rom aviRebT, darCeba 

rva}. aseTive msjeloba warimarTeba danarCen magaliTebzec. 

49/3/! sbpefopcjt! lmfcb! vnbsujwft! tbzpgbdypwsfcp! bnpdb. 

ofcTj/! amocanebi, rogorc yovelTvis, safexurebad amoixsneba. 

amocanas winaswar emateba SekiTxva imis Sesaxeb, raodenoba 

gadidda Tu Semcirda, da ramdeniT. 

1) erTxel didi Tovli movida. bavSvebma ezoSi Tovlisgan 

xuTi Tovlis papa gaakeTes. meore dRes cota daTba da ori 

Tovlis papa dadna. danarCenebi ki ar damdnara, radganac 

CrdilSi idga. moimata Tu moiklo Tovlis papebis raodenobam? 

{moiklo} ramdeniT? {oriT} ramdeni Tovlis papa darCa ezoSi? 

{sami} rogor gaige? {xuTs gamovakeli ori}. 

aseve damuSavdeba kidev ori amocana raodenobis klebaze, 

masw.-i advilad Seadgens maT (magaliTad, budidan ramdenime 

bartyi ukve dafrenilia, ramdenime qliavi SeiWama an sxva). 

Tan dafaze sqemebi ixazeba. magaliTad, daixateba xuTi 

elifsi (ès viTom Tovlis papebia~) da Semdeg ors xazebi 

gadaesmeba. 

4) yuTSi imdenive TeTri burTula ido, ramdenic Savi. mere 

amoiRes oTxi TeTri burTula. romeli gaxda meti? ramdeniT? 

ratom? 

5) dafaze exaza wreebi da kvadratebi, toli raodenobiT.

- 138 - 

mere sami wre waSales. romeli gaxda naklebi? ramdeniT? 

ratom? 

es amocanebi iseve amoixsneba, rogorc 37.2-is bolo ori 

amocana. 

49/4/!dysblvUyfefcjt!ebzpgb!tywb!nsbwbmlvUyfefcbe/! 


49/5/! `nbUfnbujlvsj! tjuzwfcjt! ebtbyfmfcb~/ moswavleebma 

rig-rigobiT unda daasaxelon is sityvebi, romlebic iswavles 

maTematikis gakveTilebze: erTi, ori, sami, ... (sadamdec 

naswavli aqvT, meti ar SeiZleba!); pirveli, meore, mesame da 

ase Semdeg; Sekreba, mimateba, gamokleba, SesaZlebloba, 

Sesabamisi, dajgufeba, zemoT, qvemoT, wvrili da ase Semdeg; 

metia, naklebia, tolia, igive, imdenive da ase Semdeg; wre, 

rgoli, wertili, monakveTi da ase Semdeg... roca yvela 

moswavlis rigi Caivlis, daiwyeba meore wre. 

magram ar iTvleba: damaxinjebulad warmoTqmuli sityva, 

sxvis mier ukve dasaxelebuli sityva da xuTi wamis (an raime 

drois _ gaaCnia, romeli wrea) Semdeg naTqvami. 

g 39. vlvUwmb/!tusjr/!Tfwtfcb-!{feb!nbskwfob!///!vksfej/ 

4:/1/! 38.6 _ SemoqmedebiTia. 38.7 saintereso amocanaa _ 

rogor moaxerxes gadaxatva. Semdeg masw.-ma dasvas SekiTxva: _ 

jvrebia meti Tu sarkmelebi? 

4:/2/! sjdywUb! sjhj;! ! fsUj-! psj-! ///! -! swb-! dysb/! ! vlvUwmb/!! 

dafaze momzadebulia TviTeuli am ricxvis `domino~, oRond 

TviT `dominoebis~ Tanmimdevroba areulia. _ es ra 

raodenobaa? (moswavlem dauTvlelad unda Tqvas). romelia 

meti _ es Tu es? es Tu es? es Tu es? romelia meti _ cxra 

Tu rva? rva Tu Svidi? Svidi Tu eqvsi? romelia naklebi _ 

eqvsi Tu cxra? rva Tu eqvsi? 

_ es raodenobani areuladaa dalagebuli. rogor davalagoT 

kargad? sul pirvelad romeli iyos? {erTeuli} gamovides 

Sorena da gadaawyos! erTeulis mere ra davawyoT? {wyvili} 

Semdegi? {sameuli} (da ase Semdeg). ratom davalageT ase?

- 139 - 

{imitom, rom jer davawyveT yvelaze patara, mere _ ufro 

didi da ase Semdeg} nana, aba daiTvale cxramde! (iTvlis). 

ratom Tqva nanam jer Svidi da mis Semdeg _ rva? {imitom, 

rom Svidi naklebia, vidre rva} ratom Tqva jer eqvsi da 

Semdeg _ Svidi? (da kidev ramdenime aseTi SekiTxva). 

_ axla daxedeT suraTs (39.1). sworadaa Tu ara dalagebuli 

raodenobani? {ki} romeli raodenobaa marcxnidan meoTxe? 

meeqvse? pirveli? mecxre? oTxeuli meramdenea? Svideuli? 

romeli raodenobaa marjvnidan meore? mesame? meeqvse? 

pirveli? mecxre? meSvide? meoTxe? rvaeuli marjvnidan 

meramdenea? marcxnidan? sameuli marcxnidan meramdenea? 

marjvnidan? (da kidev ramdenime aseTi SekiTxva). axla xatiam 

daiTvalos ukuRma, cxridan erTamde! Tan am suraTs ucqire! 

erTi-orma sxva moswavlemac daiTvalos ukuRma. 

4:/3/!sjhpcjUj!sjdywfcj;!nfTwjef-!nfswf-!nfdysf/!!qjswfmjebo! 

nfdysfnef! sjhpcjUj! sjdywfcjt! eblbwTjsfcb!obtxbwm! tjwsdjU! 

njnbsUfcfcUbo/! amas bevri dro aRar dasWirdeba. pirdapir 

ganvixilavT nakvTebis rigs (suraTi 39.2) da davsvamT amgvar 

SekiTxvebs: _ romeli nakvTia marcxnidan (an: marjvnidan) 

meSvide? mecxre? kvadrati marcxnidan meramdenea? marjvnidan? 

Semdeg ganvixilavT mTw-is ydis suraTs da movZebniT, vin 

aris meSvide, merve da mecxre. gavimeorebT, vin aris meore, 

mexuTe da ase Semdeg. 

4:/4/!gfsbej!obybufcjU!Tfwtfcvmj!dysjmjt!fsUj!tusjrpojt! 

Tfwtfcb-! :.vksfejboj! lwbesbuvmj! dysjmjtbUwjt;! {feb! 

nbskwfob-! rwfeb! nbsdyfob-! {feb! Tvb! eb! tywb! vksfefcj/ dafaze 

daxazulia carieli cxrili 3 striqoniTa da 3 svetiT. 

moswavleebi jer adgilidan daTvlian am cxrilSi striqonebis 

raodenobas {cxrilSi sami striqonia}. ramdeni svetia? {sami}. 

gamovidnen eliso da aleko! elisom ipovos am cxrilis qveda 

marcxena ujredi! alekom zeda marjvena ujredi! 

_ axla Tqvens rveulebSi daakvirdiT aseTive cariel 

cxrils (39.3). moZebneT misi zeda marjvena ujredi da 

CaxateT Sig wertilTa lamazi sameuli! moZebneT cxrilis

- 140 - 

zeda marcxena ujredi da CaxateT Sig erTi wertili! moZebneT 

cxrilis qveda marcxena ujredi da CaxateT Sig wertilTa 

lamazi Svideuli! moZebneT misi qveda marjvena ujredi da 

CaxateT Sig wertilTa lamazi cxraeuli! axla vin mixvdeba, 

ra raodenoba unda iyos zeda Sua ujredSi? {ori} ratom? 

{imitom, rom erTsa da sams Soris oria} ra raodenoba unda 

iyos qveda Sua ujredSi? {rva} ratom? {imitom, rom Svidsa da 

cxras Soris rvaa} axla, ra unda iyos Sua, meore striqonis 

ujredebSi? ra unda iyos meore striqonis marcxena ujredSi? 

{oTxi} ratom? meore striqonis marjvena ujredSi? {eqvsi} 

ratom? Sua ujredSi? {xuTi} ratom? 

Semdeg moswavleebi daaTvaliereben msr-s Sevsebul cxrils 

39.3. masw.-i yvela SekiTxvas svams mxolod sityvierad, 

miTiTebisa da gaTvalsaCinoebis gareSe. visac SekiTxvis 

gaazreba an pasuxis gacema gauWirdeba, gverdze mjdomi 

megobari miexmaros. 

_ davaTvalieroT es Sevsebuli cxrili. ramdeni striqoni 

aqvs mas? {oTxi} ramdeni sveti aqvs? {rva} cxrilSi 

striqonebi yovelTvis zemodan unda gadavTvaloT! romelia 

meore striqoni? {wiTlebisa} mesame? {ruxebisa} wiTlebis 

striqoni meramdenea? {meore} carieli striqoni? {meoTxe} 

axla vnaxoT svetebi. samkuTxedebis sveti meramdenea? {mesame} 

eqvskuTxedebisa? {meeqvse} romelia merve sveti? {rvakuTxedebisa} 

(da kidev ramdenime aseTi SekiTxva). axla moZebneT 

sul zeda marjvena ujredi. ra aris masSi? {TeTri 

rvakuTxedi}. romeli striqonia carieli? {meoTxe} modi, es 

meoTxe striqonic romelime sxva ferisa iyos. romeli feri 

SevarCioT? kargi, iyos lurji. CaxateT qveda marcxena cariel 

ujredSi lurji feri! (miuTiTebs qveda marcxena ujreds). aq 

ra feris wre unda exatos? {lurji} CaxateT! Semdeg ujredSi 

ki lurji ra unda exatos? {samkuTxedi}. CaxateT! danarCen 

ujredebs ki Sin SeavsebT (39.6)! 


gadakveTis ujredi...) jer isev araa saWiro. moswavleebi

- 141 - 

gumaniT mixvdebian, rogor Seavson carieli ujredebi Sin. 

4:/5/! `txsbgj! qbtvyfcj! `! sbpefopcbUb! Tfwtfcb~/ pirvel 

wreze raodenobas avseben Svidamde, meore wreze _ rvamde, 

mesameze da momdevnoebze _ cxramde. masw.-i akakunebs an ukravs 

taSebs, magaliTad, samjer. moswavlem Tan gonebaSi unda 

iTvalos {erTi, ori, sami} da Semdeg daukras imdeni taSi, rom 

sul gamovides Svidi taSi {oTxi, xuTi, eqvsi, Svidi}. masw.-i 

auxsnis klass, rom gonebaSi daTvlis gareSe Zneli iqneba. 

g 40. hbnfpsfcb/!sjdyw/!ljcf/!obybu/!npUbwt/!dysjmjt!vksfeTj/ 

51/1/!39.8 _ SemoqmedebiTia. 

51/2/ Ubwj! 6/2.jt! Tfkbnfcb! eb! hbnfpsfcb/! masw.-i dafasTan 

gamoZaxebul oTx moswavles daavalebs dasvan erTi, ori, sami, 

oTxi wertili _ `dominoseburad~, erT rigSi. 

meoredac oTx sxva moswavles gamoiZaxebs da daavalebs, 

dasvan xuTi, eqvsi, Svidi da rva wertili _ gagrZelebaze. 

_ ra gvaklia? {cxraeuli} gamodi da dasvi! axla 

adgilidan swrafad mipasuxeT: es ra raodenobaa? es? romela 

meti _ cxra Tu eqvsi? cxra Tu rva? cxra Tu Svidi? (da ase 

Semdeg _ pasuxebi unda iyos Zalian swrafi). 

Semdeg Sesruldeba msr-s 40.1 (romelic ukve gaferadebulia). 

_ jer davakvirdeT am rigs, qveda rigs. raodenobani 

kargadaa Tu ara dalagebuli? SeamowmeT! {ki} axla davaTvalieroT 

zeda striqoni. jer SemoxazeT sul marcxniv, nakvTebis 

wyvili! ra nakvTebia es? {rvakuTxedebia} maT Semdeg risi 

grovaa? {monakveTebisa} ramdenia? {cxra} Semdeg? {wreebis 

rvaeulia} Semdeg? {SvidkuTxedTa xuTeulia} Semdeg? {samkuTxedTa 

eqvseulia} Semdeg? {oTxkuTxedTa Svideulia} axla 

TviTeuli es grova xazebiT SevaerToT mis Sesabamis raodenobasTan 

qveda striqonidan! rasTan SevaerToT rvakuTxedebis 

wyvili? {oreulTan} SeaerTeT! (da ase Semdeg). 

51/3/! sjdywUb! ljcf! eb! njtj! tbgfyvsfcj/ am gakveTlilze 

saWiroa `dominosebri~ ricxvTa kibis saklaso suraTi (ix. 

saerTo miTiTeba II-5).

- 142 - 

jer swrafad gakeTdeba ramdenime savarjiSo “-iT metisa~ 

da “-iT naklebis~ dawyobaze (ix. 21.2 da 25.2). _ daawyve 

samze oTxiT meti kubi! daawyve eqvsze oriT naklebi wigni! 

dafaze daxate oTxze oriT naklebi rgoli! dafaze daxate 

samze oTxiT meti monakveTi!.. 

Semdeg dafaze daikideba “dominosebri~ ricxvTa kibis suraTi. 

_ ra xatia aq? {kibe} es Cveulebrivi kibea Tu ramiTaa gamorCeuli? 

ras amCnevT? {zed ricxvebia, raodenobebia} diax, ricxvebia, 

amitom hqvia am kibes ricxvTa kibe. ricxvTa kibe! aba, 

wavikiTxoT es ricxvebi! (masw.-i maCvenebel joxs gaayolebs `dominosebr~ 

ricxvebs erTidan cxramde, moswavleebi kiTxuloben. 

_ kargi, axla gelam miTxras, es meramdene safexuria? {mesame} 

manana, es? {merve} (da ase Semdeg). axla gamovides konstantine 

da kibis mecxre safexuri gviCvenos! meore? (da ase Semdeg). 

51/4/! gfsbej! obybujt! bo! oblwUjt! npUbwtfcb! obxjmpcsjw! 

Tfwtfcvmj!dysjmjt!tbUbobep!vksfeTj/!! 

dafaze winaswar unda iyos momzadebuli iseTive cxrili, 

rogoricaa msr-Si (40.3). siadvilisaTvis es cxrili dafaze 

SeiZleba ar iyos mTlianad daxazuli: iyos mxolod sami 

striqoni (meoTxe carieli striqonis gareSe) da xuTi sveti. 

mTeli momodevno msjeloba unda warimarTos dafaze daxazul 

cxrilze saTanado svetebis, striqonebisa Tu ujredebis 

miTiTebiT. Tan moswavleebi msr-s cxrilSi Seavseben. cxadia, 

Zalian sasurvelia feradi carcebis gamoyeneba. 

_ davakvirdeT cxrils. romelia misi meore striqoni? 

{ruxebisa} wiTlebis striqoni meramdenea? {mesame} axla 

vnaxoT svetebi. meore sveti risia? {rgolebisa} kvadratebis 

sveti meramdenea? {mexuTe} meoTxe risia? mesame sveti? 

_ meramdene svetia yvelaze ufro metad Sevsebuli 

nakvTebiT? {meore} axla SevavsoT misi Semdegi, mesame sveti, ai 

es. risia mesame sveti? {samkuTxedebisa} ra aris am svetis 

meore ujredSi? {ruxi samkuTxedi} ratom? {imitom, rom es 

ujredi aris ruxebis striqonSi da samkuTxedebis svetSi} ra 

unda CavxatoT amave svetis mesame, qvemoTa ujredSi? {wiTeli

- 143 - 

samkuTxedi} ratom? {imitom, rom es ujredi aris wiTlebis 

striqonSi da samkuTxedebis svetSi}. CavxatoT! 

da ase Semdeg _ Seivseba kidev ori sveti. Semdeg SeirCeva raime 

feri meoTxe striqonisaTvis, magaliTad, mwvane. msr-s cxrilis 

marcxena qveda ujredi mwvaned gaferaddeba. cxrilis mor- 

Cenili carieli ujredebi Seivseba saSinao davalebaSi (40.6). 

TfojTwob/ moswavleebs cxrilis Sevseba Tavidan SeiZleba 

gauWirdeT xolme, magram es araferia, male kargad aiTviseben. 

51/5/! `sbpefopcbUb! ebnxlsjwfcb~/! masw.-s momzadebuli aqvs 

patara muyaoebze `dominoebi~ erTidan cxramde. maT moswavleebs 

urigebs. yvelas TiTo `domino~ Sexvdeba, ramdenime 

moswavles _ erTidaigive. masw.-is taSis Semokvraze 

moswavleebi erTad unda gamovidnen da rac SeiZleba swrafad 

moewyon rigis mixedviT: jer, yvelaze marcxniv, unda dadgnen 

èrTeulebi~, Semdeg _ òreulebi~ da ase Semdeg. mowyoba 

damTavrebulad CaiTvleba, roca Sewydeba xmauri da moZraoba. 

---------------------------------------------------------------------------------- 

J.7/!bsjUnfujlvmj!ojTofcj/ Semogvaqvs sxva ariTmetikuli niSnebic: 

utolobisa da tolobis niSnebi. wers mxolod masw.-i 

(carciT dafaze), moswavleebi klasSi moZravi (magnituri) 

cifrebiT `weren~ dafaze. namdvili wera jer ar iwyeba! 

moswavleebs SeiZleba erTmaneTSi aerioT xolme da niSnebi. es 

araferia, TandaTan daimaxsovreben. miT umetes, masw.-i miaqcevinebs 

yuradRebas, rom saiTac aqvs ufro farTo mxare, iqiT unda iyos 

meti ricxvi; xolo tolobis niSans orive mxare erTnairi aqvs, 

amitom toli ricxvebi unda eweros! 

Tan xdeba naswavlis gameoreba, ukve cifrul CanawerTa doneze. 

masw.-s ar unda SeeSalos: oTxeuli esaa oTxi cali rame 

(magaliTad, oTxi wre, `dominoseburad~ an sxvagvarad ganlagebulni), 

xolo oTxiani anu 4-iani esaa Canaweri, cifri 4. 

termins `cifri~ masw.-i TviTon xmarobs saWiro dros. misi 

ganmarteba jer araa saWiro. 

---------------------------------------------------------------------------------- 

g 41. djgsfcj/!ojTboj >.!oblwUjt!npUbwt/!dysjmjt!vksfeTj/

- 144 - 

52/2/!djgsfcj!2-!3-!4/!!nfupcjt!ojTboj >.!vupmpcbUb!xbljUywb;! 

2>1, 3>2, 3>1. mTw gadaiSleba `1,2,3~-ze. 

_ ramdeni wrea aq? {erTi} miCveneT erTi TiTi, erTi xeli, 

erTxel daakakuneT! erTi ra aqvs adamians? ricxvi erTi ase 

iwereba (miuTiTebs mTw-Si). esaa erTiani! 

_ ramdeni wrea aq? {ori} miCveneT ori TiTi, ori xeli, 

orjer daakakuneT! ori ra aqvs adamians? ricxvi ori ase 

iwereba (miuTiTebs mTw-Si). esaa oriani! 

aseve moviqceviT samzec. 

_ miCvene 1-iani! miCvene 2-iani! miCvene 3-iani! miCvene wreebis 

sameuli! miCvene oriani! miCvene wreebis wyvili! miCvene 

erTiani! 

_ romelia meti, ori Tu erTi? metoba ase aRiniSneba: >. 

magaliTad: 2 > 1 _ es ase ikiTxeba: ori (masw.-i daadebs 

TiTs) metia (daadebs TiTs) erTze (daadebs TiTs). axla me 

dagiwerT da Tqven waikiTxeT: 3 > 2, 3 > 1. 

_ axla vnaxoT suraTi (41.1). ra ricxvebi weria aq? {erTi, 

ori sami} TviTeul cifrs qveS miuxateT misi Sesabamisi raodenobis 

wertilebi. lamazad ganlagebuli wertilebi. ra unda 

davxatoT erTianis qveS? {erTeuli, erTi wertili} dasviT! 

orianis qveS? {ori wertili} dasviT! samianis qveS? {sameuli, 

sami wertili} dasviT, lamazi ganlagebiT! (masw.-i Seamowmebs, 

wertilTa sameuli rom `dominoseburad~ iyos). axla naxatebi 

xazebiT SeaerTeT Sesabamis cifrTan. yvelaze marcxniv ra 

xatia? {ori Taguna} romel cifrTan unda SevaerToT Tagunebis 

wyvili? {orianTan} SeaerTeT! Semdegi ra aris? {katebis 

sameuli}... Semdegi? {matli} rasTan SevaerToT? {erTianTan}. 

(da ase Semdeg _ matlic SeerTdeba erTianTan). axla ki wavikiTxoT, 

romeli metobani weria lurj CarCoSi. pirveli 

waikiTxos avTom, mxolod (am sityvis gamokveTiT) pirveli! 

{sami metia orze} axla paatam waikiTxos meore metoba! (da ase 

Semdeg). axla davakvirdeT am cifrebs. samianis zemoT erT 

ujredSi ra aris? {wyvili} romeli raodenoba unda iyos 

meore, cariel ujredSi? sami ris tolia: orisa da ... ris?

- 145 - 

{erTis} diax, erTi. erTeuli unda iyos. dasviT! (aseve Seivseba 

danarCeni ori carieli ujredic). 

52/3/!sjdywjt!nbufcb!sjdywUb!ljcf{f!{fwjUlfo!xbobdwmfcjtbt/! 

_ ramdeni sarTuli aqvs am saxls? (dafaze sqematurad xatia 

9-sarTuliani saxli). {cxra} davaweroT! (`daawers~ gverdidan 

dominosebr ricxvebs erTidan cxramde). saxlSi maRla 

asvlisas (xeliT uCvenebs moZraobas) sarTulis ricxvebi 

imatebs Tu iklebs? {imatebs} diax, jer pirveli sarTulia, 

Semdeg... romeli? {meore} ai, oreulia. Semdeg? {mesame}... ai, 

sameulia (da ase Semdeg). qvemoT Camosvlisas (kvlav xeliT 

uCvenebs) rogor icvleba es ricxvebi? {iklebs} jer mecxrea, 

Semdeg... romeli? {merve} ... ai, rvaeulia. 

_ daakvirdiT axla ricxvTa kibis safexurebs. ai es daTunia 

amyavs kibeze zemoT. rogor icvleba ricxvebi _ imatebs Tu 

iklebs? {imatebs} qvemoTken Camosvlisas? {ricxvebi iklebs}. 

rac ufro maRlaa daTunia, miT ufro didi ricxvia, rac ufro 

dablaa, miT ufro... ra? {patara ricxvia}. aba axla romel 

safexurzea daTunia? {mesameze} aviyvanoT zeviT, ricxvebi imatebs, 

axla romelzea? {merveze} axla qveviT CamoviyvanoT, 

ricxvebi iklebs, axla romel safexurzea daTunia? {meoreze} 

(da ase Semdeg). 

52/4/! gfsbej! oblwUjt! npUbwtfcb! dbsjfmj! dysjmjt! tbUbobep! 

vksfeTj/!yvelaferi iseve, rogorc 40.3-Si. Seivseba ramdenime 

sveti, ramdenic moeswreba, danarCenebi _ saSinao davalebad. 

52/5/! `txsbgj! qbtvyfcj! `! tbqjsjtqjspoj~/ ix. 12.4. masw.-i 

ambobs romelime sityvas naswavl raodenobriv da sivrciT 

mimarTebaTagan: bevri, mravali, cota, meti, naklebi, didi, 

patara, maRali, dabali, zemo, zemoT, zeda, qvemo, qvemoT, 

qveda, wina, win, ukana, ukan, SigniT, Siga, gareT, gare, msxvili, 

wvrili, sqeli, Txeli, farTo, ganieri, viwro. 

emateba axlebi: marjvena, marjvniv, marcxena, marcxniv, 

metobis niSani, naklebobis niSani, Sekreba, gamokleba. 


amaTi sapirispiroebi Znelia. amitom eseni meore wreze unda

- 146 - 





g 42. djgsfcj/!!vupm/!xbljUywb/!oblwUjt!npUbwtfcb!dysjmTj/ 

53/2/!djgsfcj!5-!6-!7/!!vupmpcbUb!xbljUywb: 5>4, 6>5, 6>4, 5>2 

eb! tywb/! yvelaferi keTdeba iseve, rogorc 41.1-Si, oRond > 

niSani ukve naswavlia. garda amisa, naxatebis grovaTa SeerTeba 

Sesabamis cifrTan gatanilia calke amocanad saSinao 

davalebisaTvis (42.8). 

53/3/!sjdywjt!.jU!hbejefcb!eb!sjdywUb!ljcf{f!tbUbobep!xbobd. 

wmfcb/!jer swrafad vimeorebT naswavl sakiTxs _ `-iT metoba~. 

dafaze `dominoseburi~ xuTeuli da oTxeulia, erTi moswavle 

daxatavs xuTeulze samiT met wres, meore _ oTxeulze. 

Semdeg erTi mcire siaxle Semogvaqvs _ sawyisi raodenoba 

mocemulia ara TvalsaCinod, aramed cifris saxiT. 

masw.-i dafaze wers 2. _ vin daxatavs amaze samiT met 

rgols? {moswavle daxatavs, masw.-i dawers 5). rogor mixvdi, 

rom xuTi rgoli unda dagexata? {ors mivumateT sami} ratom 

miumate ors sami? {3-iT metio, ese igi, imdenive, ori da kidev 

sami, amitom SevkribeT}. diax, orze samiT metia xuTi (da 

dawers: 2 + 3 = 5). 

_ axla ciram ricxvTa kibeze gviCvenos meore safexuri! aba 

es ori gaadide samiT! ricxvis gasadideblad, mosamateblad ra 

unda qna, saiT unda waxvide? {zemoTken} wadi zemoTken! magram 

ramdeni safexuriT unda axvide? ramdeniT unda gavadidoT 

ori? {samiT} hoda adi sami safexuriT zemoT! ra ricxvi 

miiRe? {xuTi} diax, orze samiT meti _ xuTia! 

aseve sxva moswavleebic gaakeTeben ramdenime savarjiSos. 

53/4/! gfsbej! oblwUjt! npUbwtfcb!dbsjfmj!dysjmjt!tbUbobep! 

vksfeTj!tjuzwjfsj!bSxfsjt!Tfnefh/!yvelaferi iseve, rogorc 

41.3-Si, oRond SekiTxvebi da pasuxebi asea: _ gaayoleT Tvali 

wreebis striqons. moZebneT am striqonis gadakveTa wiTlebis

- 147 - 

svetTan! striqonis romeli ujredia? {meore} ra unda exazos 

masSi? {wiTeli wre} CaxateT! axla miTxariT, romel ujredSi 

unda exazos ruxi xuTkuTxedi? {xuTkuTxedebis striqonisa da 

ruxebis svetis gadakveTaze}. 

ase Seivseba cxrili. cariel striqonsa da cariel svets ki 

Sin Seavseben. TviTon moigoneben axal nakvTsa da axal fers. 

masw.-i yuradRebas miaqcevinebs: _ am cxrilSi rame 

sxvagvarad xom ar aris, vidre adre naswavl cxrilebSi iyo? 

{svetebi ferebisaTvisaa, xolo striqonebi _ nakvTebisaTvis, 

piriqiTaa} diax, cxrilSi asec SeiZleba da isec. rogorc 

gvsurs. orivegvarad SeiZleba. 

53/5/! `txsbgj! qbtvyfcj! `! lblvo.ubTfcj~/ ix. 18.4, oRond 

raodenoba aRwevs cxras (nacvlad eqvsisa). 

g 43. djgsfcj/!ojTboj =.!njnbufcb!sjdywUb!ljcf{f/!hbnfpsfcb/ 

54/2/!djgsfcj!8-!9-!:/!!upmpcjt!ojTboj =.!upmpcbUb!xbljUywb;! 

88-!99-!::

- 148 - 

aviTarebs III Tavis sakiTxebs, arsebiTad axali araferia. 

ricxvTa kibe (daxazuli) Zalian kargi TvalsaCinoebaa Sekrebisa 

da gamoklebisaTvis. momdevno klasebSi igi ricxvTa sxivad 

gardaiqceva. saTanado sagakveTilo monakveTebi Zalian advilebia, 

maT mcire dro eyofa. 

pirveli sami gakveTilis ganmavlobaSi saWiroa ricxvTa kibis 

iseTi saklaso suraTi, romelsac aweria ara cifrebi, aramed 

`dominosebri ricxvebi~. jer erTi, cifrebi jer ara aqvT 

naswavli, meorec, ase jobia: moswavleebi xom amgvar ricxvebs arian 

SeCveulni. ricxvTa kibis Tema maTTvis unda iyos uSualo 

gagrZeleba 39.1-isa: ricxvTa rigi erTi, ori, sami, ... , cxra; Tvla 

da ukuTvla. ubralod, es raodenobani ganlagebulia kibis 

safexurebze. 

Tuki ver moxerxda saWiro saklaso TvalsaCinoebis momzadeba, 

masw.-s advili gamosavali aqvs: Cveulebriv ricxvTa kibis suraTze 

(saklaso plakatze) cifrebs droebiT sakrobiT gadaewebos 

baraTebi saTanado raodenobaTa dominosebri ganlagebiT. sami 

gakveTilis Semdeg es baraTebi moixsneba da darCeba Cveulebrivi 

ricxvTa kibe. 

---------------------------------------------------------------------------------- 

54/3/! sjdywjtUwjt! sjdywjt! njnbufcb! eb! sjdywUb! ljcf{f! 

tbUbobep!xbobdwmfcb/!am gakveTilze ukve Cveulebrivi ricxvTa 

kibe Semogvaqvs. saklaso oTaxSic misi suraTi kidia, Tan 

moswavleebs gadaSlili aqvT mTw ricxvTa kibis suraTze. _ am 

ricxvTa kibeze ra siaxlea? ra aris Secvlili? {ricxvebi 

axleburad weria} diax, ricxvebi weria cifrebiT, xom ukve 

naswavli gvaqvs es cifrebi! aba, waikiTxeT, romel safexurzea 

daTunia? axla? 

Tan gavimeorebT ricxvis -iT gadidebas ricxvTa kibeze 

saTanado wanacvlebasTan dakavSirebiT (ix. 42.2). Semdeg 

gavakeTebT imave savarjiSoebs, oRond ricxvis `-iT gadidebis~ 

nacvlad iqneba ricxvisTvis ricxvis mimateba. masw.-i dafaze 

dawers Sekrebis magaliTs, magaliTad, 5 + 3, moswavle ki mas 

ricxvTa kibeze asrulebs.

- 149 - 


carieli cxrili 43.3. moswavleebi jer daTvlian, ramdeni 

striqoni da ramdeni sveti aqvs cxrils. Semdeg Tavad 

SearCeven or fers striqonebisaTvis (oRond yvelas 

erTidaigive ferebi unda hqondes). _ svetebi ki nakvTebisaTvis 

iyos. ramdeni sxvadasxva nakvTi dagvWirdeba? {eqvsi} SevarCioT 

nakvTi meore svetisaTvis. kargi, iyos kvadrati. CaxateT zeda 

ujredSi! ar SegeSaloT, xom ar aferadebT? pirvel striqonSi 

rogori nakvTebi unda exatos? {TeTri, gauferadebeli} 

CaxateT kvadarati! amave (am sityvis gamokveTiT) svetis meore 

ujredSi ra unda exazos? {mwvane kvadrati} CaxateT! (da ase 

Semdeg _ Seivseba oriode sveti, danarCenebi _ Sin). 

54/5/ masw.-s xelSi cifrebiani (1-dan 9-mde, rac aqvT naswavli) 

baraTebis dasta ukavia da rig-rigobiT, sxvadasxva moswavles 

myisierad uCvenebs gverdiT gadabrunebul (`dawvenil~) 

romelime cifrs. moswavlem unda daasaxelos es cifri. meore 

wreze _ am cifris momdevno anu Semdegi cifri; mesame wreze 

_ wina cifri da ase Semdeg: wina, momdevno, wina, momdevno... 

g 44. ojTboj!/!vupm/!xbljUywb/!ebvysfm/!ebyb{/!nbsUlvUy/ 

55/2/!oblmfcpcjt!ojTboj!/!vupmpcbUb!xbljUywb;!1 6. _ vin 

waikiTxavs? (moswavle gamova da kiTxulobs, Tan TiTs adebs 

saTanado niSans). daakvirdiT metobis niSnis gaxsnil, farTo 

mxares. romel mxaresaa farTo? {marcxniv} viwro? {marjvniv} 

romeli ricxvia meti _ farTo mxarisken Tu wverosaken? 

{farTosken} diax, mxarec ufro farToa da ricxvic ufro 

didia. saiTac farTo mxarea, iqiT weria didi ricxvi! 

Semdeg masw.-i dafaze dawers: 4 = 4, 7 = 7. _ vin 

waikiTxavs? es ra niSania? {tolobisa} am niSans saiT aqvs

- 150 - 

farTo mxare? {arsaiT, orive mxare tolia} ratomaa ase? 

{imitom, rom tolobis niSania, ricxvebi xom tolia!} diax, 

mxareebic tolia da ricxvebic tolia! 

Semdeg masw.-i dafaze dawers: 1 < 3, 5 < 9, 2 < 6. 

_ Tqveni azriT, es rogor wavikiTxoT? daakvirdiT am niSans! 

saiT aqvs mas farTo mxare? {marjvniv} metobis niSans SexedeT, 

mas saiT aqvs farTo mxare? {marcxniv} diax, am niSans piriqiT 

aqvs. aba es ra niSani iqneba? ra aris metobis sapirispiro? 

{naklebi} metobis niSnis sapirispiro raRa iqneba? {naklebobis 

niSani} amas vin waikiT-xavs? {erTi naklebia samze, xuTi 

naklebia... _ masw.-i Tan TiTs adebs saTanado niSnebs} diax, esaa 

naklebobis niSani. saiTac mas aqvs farTo mxare, iqiT ufro 

didi ricxvi weria Tu ufro patara? {ufro didi} viwro 

mxarisken, wverosken? {ufro patara}. axla wavikiTxoT es 

utolobebi (msr-Si 44.1). daiwyebs amirani! 

55/3/! Tflsfcjt! Tftsvmfcb! sjdywUb! ljcjt! nfTwfpcjU/! keTdeba 

igive savarjiSoebi, rac 43.2-Si, oRond erTi siaxliT: masw.-i 

mTel klass aZlevs Sekrebis SedarebiT Znel magaliTs (5+4, 

3+5, 2+6 da sxva), moswavleebi ki am Sekrebas asruleben 

ricxvTa kibeze mTw-Si, pasuxs zepirad amboben. 

---------------------------------------------------------------------------------- 

JJJ.5/! nbsUlvUyfej-! njtj! ebyb{wb/! nsbwbmlvUyfejt! )kfs! bn! 

ufsnjojt!hbsfTf*!hwfsefcj!eb!lvUyffcj-!nbUj!ebUwmb/! 

iseve rogorc yvela sxva nakvTi, marTkuTxedic Semogvaqvs 

mxoloddamxolod TvalsaCino CvenebiT, yovelgvar ganmartebaTa 

gareSe. sazogadod, gansazRvrebani, magaliTad, kuTxisa an marTkuTxedisa, 

dawyebiT klasebSi sruliad uadgiloa. maT bavSvi mainc ver 

gaiazrebs da kaWkaWiviT unda imeoros. ufro metic, sakmaod Zneli 

gasaazrebelia isic ki, Tu sazogadod ras niSnavs sityvieri 

gansazRvrebis mixedviT moqmedeba. 

moswavleebs unda vaswavloT nakvTebis cnoba (Tanac, 

aucileblad yovelgvar mdebareobaSi!) da maTi daxazva Tu gazomva. 

garda amisa, viwyebT kanonzomierebis Temas. iqneba jer Zalian 

advili amocanebi logikuri kanonzomierebis darRvevaze romeli

- 151 - 

arRvevs kanonzomierebas; Semdeg nakluli kanonzomierebis 

Sevsebaze. termini `kanonzomiereba~ amgvar amocanebSi gulisxmobs 

imas, rom yvela wevri raRaciT erTmaneTs hgavs, raRarc saerTo 

aqvT, garda erTisa. swored es erTia gamonaklisi, anu zedmeti, 

anu kanonzomierebis damrRvevi. meoregvar amocanebSi ki mocemulia 

wevrTa rigi, romlebsac raRac kanonzomiereba aerTianebs. 

gamonaklisi araa, magram saWiroa am kanonzomierebis mixedviT 

rigis gagrZeleba, carieli adgilisa Tu ? niSnis Sevseba. 

xSirad amgvar amocanebSi araa sakmarisi marTebuli pasuxis 

moZebna. mas bavSvic advilad poulobs xolme gumaniT. saWiroa 

dasabuTebac: Tu ratomaa es pasuxi marTebuli. 

termins `kanonzomiereba~ ver gaveqceviT, masw.-i Tavidan ixmars 

yovelgvari axsna-ganmartebis gareSe, saWiroebis mixedviT, magram 

yovelTvis konkretul da TvalsaCino viTarebaSi. moswavleebi am 

sityvis mniSvnelobas mainc ver Caswvdebian, pirvel klasSi es arcaa 

saWiro. ubralod, maT unda icodnen, ra moeTxovebaT, roca 

amocanis pirobaSi gvxvdeba es sityva. amas ki sakmaod male 

aiTviseben. sxvaTa Soris, amdagvarive viTarebaa agreTve terminebTan 

`SesaZlebloba~ da `dajgufeba~. 

---------------------------------------------------------------------------------- 

55/4/!ebvysfmbe!ebyb{vmj!nbsUlvUyfej/!dafaze erT rigSi 

dauxrelad daxazulia ori sxvadasxva marTkuTxedi, Semdeg _ 

trapeciis magvari oTxkuTxedi, Semdeg _ kvadrati. _ es nakvTi 

ramdenkuTxedia? {oTxkuTxedi} iamze, daTvale misi gverdebi! 

amgvar nakvTebs marTkuTxedi hqvia. ai es meore ra nakvTia? {esec 

marTkuTxedia} es nakvTi rigSi meramdenea {mesame} ramdenkuTxedia? 

{oTxkuTxedi} iamze, daTvale misi gverdebi! aris 

Tu ara es marTkuTxedi? {ara} ratom? {ai es adgili cerad 

aqvs, marTkuTxeds ki gverdebi yvelgan sworad unda hqondes}. 

es nakvTi rigSi meramdenea? {meoTxea} ra nakvTia? {kvadratia} 

diax, kvadratia, magram Tan marTkuTxedicaa. xom yvelgan sworad 

aqvs gverdebi! amitom kvadrati Tan marTkuTxedia! axla 

daxedeT suraTs (44.3) es wiTeli nakvTi ra aris? {oTxkuTxedia, 

marTkuTxedi} axla daxazeT kidev erTi marTkuTxedi

- 152 - 

(xazaven da aferadeben). Sin, cariel adgilas TviTon daxataven 

gverdiTa marTkuTxedismagvar axal marTkuTxedsac, danarCenebsac 

daasruleben. 

55/5/! èbnbmvmj! djgsjt! [fcob~/ is moswavle, romelmac 

kargad upasuxa, gamodis, zurgSeqceviT dgeba da Tvalebs 

xelebiT ifaravs. masw.-i erTerT moZrav cifrs malavs romelime 

merxSi an merxze. moswavle Tvalebs gaaxels da iwyebs 

damaluli cifris Zebnas. mas rig-rigobiT aswavlian gzas: _ 

gadadgi ori nabiji win! erTi nabiji marjvniv! xeli marcxniv 

gawie! cota zemoT! (da ase Semdeg). 

Semdeg malaven sxva cifrs da ase Semdeg. 

55/8/!Tuki gauWirdebaT, dafaze daixazos didi xuTkuTxedi 

da moswavlem scados misi sxvadasxvagvari gayofa. saWiro 

iqneba misTvis an diagonaliT samkuTxedis CamoWra, anda mxolod 

patara wveris waWra. 

g 45. ojTboj +/!upmpc/!xbljUywb/!nbsUlvUy/!ebvysfm/!ebyb{wb/ 


56/2/!Tflsfcjt!ojTboj/!upmpcbUb!xbljUywb;!1+1 = 2, 3+6 = 9 ! 

eb! tywb/! dafasTan ramadenime moswavle gamova. erTi Sekrebs 

(SeaerTebs) winaswar momzadebul grovebs, meore moswavle 

Sekrebs dafaze `dominoseburad~ dalagebul sameulsa da 

xuTeuls, mesamec rame amdagvars. 

masw.-i dafaze daxazavs elifss da Sig Casvams `dominoseburad~ 

dalagebul sameulsa da xuTeuls (rogorcaa mTw- 

Si). _ aq sul ramdeni wrea? {rva} davaweroT! (da elifss 

zemodan daawers did rvians). sams mivumatoT xuTi (Tan TiTs 

adebs!) tolia rvisa. es mokled ase Caiwereba (wers dafaze 

msxvilad, didi SualedebiT, elifsis qvemoT): 3 + 5 = 8. 

wavikiTxoT: sams (daadebs TiTs) mimatebuli (daadebs TiTs) 

rva (daadebs) tolia (daadebs) rvisa (daadebs). es risi niSania? 

{tolobisa} es ki risi niSania? {Sekrebisa, anu mimatebisa}. 

diax, Sekrebis. erTi patara monakveTi (dafaze daxazavs 

Sveul monakveTs) da mas damatebuli meore patara monakveTi

- 153 - 

(daxazavs meores, ise, rom jvars miiRebs). monakveTebi TiTqos 

Seikriba, ara?! amitomaa es niSani _ Sekrebis niSani. axla 

rveulSi am wiTel CarCoSi wavikiTxoT tolobebi! (msr-Si 

45.1 _ rig-rigobiT kiTxuloben, Tan TiTs ayoleben). 

56/3/!sjdywjt!lmfcb!sjdywUb!ljcf{f!rwfwjUlfo!xbobdwmfcjtbt/! 

jer swrafad gavimeorebT, rogor icvleba ricxvi ricxvTa 

kibeze zeviT wanacvlebisas. 

Semdeg yvelaferi gakeTdeba iseve, rogorc 41.2-Si, oRond 

zeviT wanacvlebis nacvlad iqneba qveviT wanacvleba da, 

Sesabamisad, ricxvis kleba. 

56/4/! nbsUlvUyfejt! ebvysfmbe! ebyb{wb! vksfefcjbo! gvs. 

dfm{f! bo! nbsUlvUybywsfmjboj! tbyb{bwjU/! masw.-i klass 

aswavlis, ujredebian badeze rogor daxazon marTkuTxedi: 

gadaTvalon ramdenime ujredi, moniSnon erTi gverdis boloebi, 

Semdeg zemoT gadaTvalon ramdenime ujredi da ase Semdeg. 

moswavleebi xazaven 45.3 badeze, daxazuls kidec aferadeben. 

daixazeba ramdenime sxvadasxva zomebis mqone marTkuTxedi. 

bolos, erT marTkuTxeds daxazaven marTkuTxaxvreliani 

saxazaviT. 

56/5/! `txsbgj! qbtvyfcj! `! nfsbnefof~/ SekiTxvebi amgvaria: 

romel sarTulze cxovrobs is, visi zemoTa (an: qvemoTa) 

mezobeli cxovrobs meSvide sarTulze? 

arcerTi sarTuli ar unda gascdes mecxre sarTuls. meore 

wreze SekiTxvebi icvleba. masw.-i dafaze daxatavs cotaTi 

mixveul-moxveul gzas da mis cal mxares _ cxra saxls. 

saxlebi isea, rom maTgan marjvena / marcxena emTxveva 

mayureblis marjvena / marcxenas. saxlebs daawers (marcxnidan 

marjvniv): 1, 2, ... , 9. SekiTxvebi amgvaria: _ romel 

nomer saxlSi cxovrobs is, visi wina (an: Semdegi) mezobeli 

cxovrobs mexuTe saxlSi? 

mesame da momdevno wreebze SekiTxvebSi gamoTqma `wina 

(Semdegi) mezobeli~ Seicvleba gamoTqmiT `marcxena / marjvena 

mezobeli~.

- 154 - 

g 46. vupm/!xbljUywb/!nbsUlvUy/!tywbebtywb!nefcbsfpcbTj/ 

57/2/!vupmpcbUb!xbljUywb: 3+1 > 3, 4+2 > 4, 5 > 2+2, 4+1 < 7 

eb! tywb/! jer swrafad gavameorebinebT Sekrebis magaliTebis 

wakiTxvas (rogorc iyo 45.1-Si). Semdeg mTw gadiSleba `9~-ze. 

_ aq ramdeni wrea? {cxra} aq wreebi rogoraa dalagebuli? 

{xuTi da oTxi} maSasadame, ramdenia xuTs mivumatoT oTxi? 

{cxra} axla SevadaroT xuTs miumatoT oTxi da eqvsi. 

romelia meti? {xuTs mivumatoT oTxi} ratom? {imitom, rom 

igi cxris tolia, cxra ki metia, vidre eqvsi}. davweroT! 

(masw.-i dafaze dawers: 5 + 4 > 6). axla es CavweroT naklebobis 

niSniT! rogor CavweroT? {6 < 5 +4}. waikiTxeT! 

_ axla daakvirdiT metobis niSansa da naklebobis naSans. 

miTxariT, wveri ufro didi ricxvisaken aris mimarTuli Tu 

ufro patara ricxvisaken? {patarasaken} metobis niSanSia ase 

Tu naklebobis niSanSi? {oriveSi asea!} diax, sul erTia, romeli 

niSania, wveri yovelTvis pataraskenaa, imitom, rom wveri 

TviTonaa patara da viwro! farTo mxare? (iseve gameordeba). 

bolos ki rig-rigobiT wavakiTxebT utolobebs wiTel 

CarCoSi 46.1. Tan TiTi gaayolon. 

57/3/! sjdywjt! .jU! Tfndjsfcb! eb! sjdywUb! ljcf{f! tbUbobep! 

xbobdwmfcb/! jer swrafad gavimeorebT, ricxvTa kibeze rogor 

icvleba ricxvis mdebareoba am ricxvis `-iT gadidebisas~. 

Semdeg masw.-i dawers: 9. _ vin dawers 9-ze 4-iT nakleb 

ricxvs? 

moswavleTa umetesobis pasuxi marTebuli iqneba. masw.-i 

moiTxovs msjelobas, rogor mixvdnen. Semdeg: _ raki xuTi 

oTxiT naklebia, vidre cxra, amitom cxra oTxiT... ra? {oTxiT 

metia, vidre xuTi). da aseve piriqiTac. 


`-iT gadidebis~ nacvlad iqneba `-iT Semcireba~ da, Sesabamisad, 

ricxvTa kibeze qveviTken wanacvleba. 

57/4/!nbsUlvUyfej!tywbebtywb!nefcbsfpcbTj/! 

magnitur dafaze ramdenime oTxkuTxedia, maTgan erTi _ 

didi wiTeli marTkuTxedi, dauxrelad mikruli, erTi _ sxva

- 155 - 

feris aseTive marTkuTxedi, erTi ki _ sxva feris 

marTkuTxedi, oRond cerad mikruli. _ es ra nakvTia? 

{oTxkuTxedi} marTkuTxedi Tu aris? {ara} ratom? {es adgili 

sworad ara aqvs} es wiTeli ra nakvTia? {marTkuTxedi} diax. 

axla SemovatrialoT! axla ra nakvTia? {isev marTkuTxedia} 

aba ra! SemotrialebiT xom ar Seicvleboda! axla vin daxatavs 

dafaze carciT or kvadrats: erTi iyos dauxrelad, sworad 

daxazuli, meore ki _ daxrilad, cerad daxazuli (gamova 

moswavle da xeliT daxatavs). axla vin daxatavs am wiTel 

marTkuTxeds, oRond ase cerad? (gamova moswavle da 

gadaxatavs). diax. es ra nakvTia? {oTxkuTxedi} marTkuTxedi 

Tu aris? {ara} ratom? {es adgili sworad ara aqvs} es ra 

nakvTia? {oTxkuTxedi} marTkuTxedi Tu aris? {ki} es? {esec 

marTkuTxedia, oRond ceradaa, daxrilia}. ai, SeiZleba 

gavasworoT! magram iyos isev ise, cerad, rogorc iyo. 

_ axla daaTvaliereT suraTi (46.3). ra aris? {saxli} diax, 

saxlia, Sedgenilia nakvTebisagan. gverdiT ki igive nakvTebia, 

oRond daSliladaa, cal-calkea. axla daakvirdiT da gaaferadeT 

sxvadasxva ferebad, oRond erTnairi nakvTebi erT ferad unda 

gaaferadoT! saxlis naxatSic da gverdiT, calkec _ erTi ferisa 

unda iyos! daiwyeT, jer marTkuTxedebi gaaferadeT! 

moswavleebi aferadeben. masw.-i, rogorc yovelTvis aseT 

dros, merxebs Soris dadis da amowmebs: xom ar gamorCaT 

daxrilad daxazuli marTkuTxedebi. _ sul ramdeni 

marTkuTxedi hqonia saxls? {rva}. 

rasac ver moaswreben, Sin daasrulon. 

57/5/! `nbUfnbujlvsj! tjuzwfcjt! ebtbyfmfcb~/ ix. 38.4. 

daemateba axali sityvebic: marTkuTxedi, cerad, metoba, 

nakleboba, ricxvTa kibe da sxva. 

g 47. ojTboj!–/!upmpcbUb!xbljUywb/!bnpdbo/!lbopo{pnjfsfcb{f/ 


58/2/!hbnplmfcjt!ojTboj!–/!upmpcbUb!xbljUywb: 9–2 = 7, 8–1 

= 7, 4–2 = 2 ...! mTw gadaiSleba `8~-ze. _ ramdeni wertilia

- 156 - 

aq? {rva} rogoraa aq dawyobili rva wre? {xuTi da sami} rvas 

rom xuTi gamovakloT, ra dagvrCeba? {sami} (masw.-i dafaze 

daxazavs elifss, zed daawers did rvians, Sig ki Casvams 

xuTeulsa da sameuls, rogorc mTw-Sia). maS axla rvas 

gamovakloT xuTi. gamovakloT! (da waSlis xuTeuls). ra 

dagvrCa? {sami} diax, rvas gamoklebuli xuTi tolia samisa 

(da elifsis qvemoT dawers: 8 _ 5 = 3). vin mixvdeba, es 

niSani ra niSania? {gamoklebisa} diax, esaa gamoklebis niSani. 

ra aris gamoklebis sapirispiro? {Sekreba} Sekrebis niSnis 

sapirispiro? {gamoklebis niSani} maS, sul radara niSnebi 

viswavleT Cven? (moswavleebi CamoTvlian, masw.-i dafaze wers > 

< = + da _ niSnebs). 

_ axla wavikiTxoT tolobani wiTel CarCoSi (47.1). 

58/3/! sjdywjtbUwjt! sjdywjt! hbnplmfcb! eb! sjdywUb! ljcf{f! 

tbUbobep! xbobdwmfcb/! jer swrafad gavimeorebT, rogor 

icvleba ricxvis mdebareoba mimatebisas (43.2). 


mimatebis nacvlad iqneba gamokleba. 

58/4/! qjswfmj-! vbewjmftj! bnpdbofcj! mphjlvsj! lbopo{pnjf. 

sfcjt! ebsSwfwb{f! `! spnfmj! bsSwfwt! lbopo{pnjfsfcbt/! jer 

swrafad gavimeorebT cerad daxazuli marTkuTxedis cnobas 

(dafaze), mis gamorCevas sxva oTxkuTxedebisagan, aramarTkuTxedebisagan. 

Semdeg masw.-i gamoiyvans xuT gogonasa da erT biWunas (an 

piriqiT). daayenebs maT areulad. _ aq erTi bavSvi arRvevs 

kanonzomierebas. yvela bavSvi raRaciT erTmaneTs hgavs, is 

erTi ki danarCenebs ar hgavs! romelia? ratom? 

_ axla davakvirdeT suraTs (47.3). ramdeni nakvTia daxazuli? 

{xuTi} am xuTidan erTi arRvevs kanonzomierebas. es erTi 

sxvebisagan sul gansxvavebulia. romelia? {meore, wre} ratom? 

{imitom, rom yvela danarCeni _ marTkuTxedebia, es erTi ki _ 

wrea}. SemoxazeT is, romelic arRvevs kanonzomierebas! 

Tuki moswavleebi Seecdebian, feris mixedviT ipovon 

gamonaklisi nakvTi, scadon, TviTon darwmundebian, rom arafe-

- 157 - 

ri gamouvaT. saqme isaa, rom kanonzomierebas mxolod erTi 

unda arRvevdes da yvela danarCeni ki raRac TvalsazrisiT 

erTnairi unda iyos. 

_ axla davakvirdeT marjvniv am raodenobebsa da ricxvebs. 

romeli arRvevs kanonzomierebas? {oTxiani, cifri} ratom? 

{imitom, rom yvela danarCeni `dominosaviTaa~ daxatuli 

wreebiT, es ki cifriTaa Cawerili}. 

58/5/!`txsbgj! qbtvyfcj~!`! oblwUfcjt! dopcb/ am TamaSisaTvis 

masw.-s winaswar unda hqondes momzadebuli muyaosgan 

gamoWrili sxvadasxvagvari da sxvadasxva feris wreebi, 

rgolebi, kvadratebi, samkuTxedebi, oTxkuTxedebi, xuTkuTxedebi, 

eqvskuTxedebi, SvidkuTxedebi da ase Semdeg _ yvela 

naswavli nakvTi. masw.-i moswavles wamierad uCvenebs nakvTs, 

Semdeg xeliT faravs an parkSi malavs. moswavlem unda Tqvas, 

romeli nakvTi iyo. 

rac ufro Zneli sacnobia nakvTi (sxvadasxvagvari xuTkuTxedebi, 

eqvskuTxedebi da ase Semdeg), miT ufro met xans 

vuCvenebT nakvTs. advilad sacnobebs (wre, rgoli, samkuTxedebi) 

ki vuCvenebT myisierad da isic mxolod pirvel wreze. 

g 48. vupm/!xbljUywb/!bnpd/!oblmvm!lbopo{pnjfsfcb{f/! 

59/1/ 47.8 _ SemoqmedebiTia. 

59/2/!vupmpcbUb!xbljUywb: 7–2 < 7, 5–4 < 9, 6 > 9–6, 8–1 > 3 

eb!tywb/!!yvelaferi keTdeba imdagvaradve, rogorc 46.1-Si. 

59/3/! hbnplmfcjt! Tftsvmfcb! sjdywUb! ljcjt! nfTwfpcjU/! jer 

swrafad gavimeorebT, rogor sruldeba ricxvTa kibeze Sekreba 

(44.2). Semdeg yvelaferi gakeTdeba iseve, rogorc 44.2-Si, 

oRond Sekrebis nacvlad iqneba gamokleba da, Sesabamisad, 

qveviTken wanacvleba. 

59/4/! qjswfmj-! vbewjmftj! bnpdbofcj! oblmvmj! lbopo{pnjf. 

sfcjt!Tfwtfcb{f/!!`{feb!nbskwfob~-!`rwfeb!Tvb~!eb!tywb!vksf. 

efcj/! ganvixilavT cxrils 48.3. jer gairkveva, ramdeni sveti, 

ramdeni striqoni aqvs mas; ra weria zeda marcxena, Sua marcxena 

da ase Semdeg ujredebSi; ra weria meore svetis pirvel

- 158 - 

Tu mesame striqonis mesame ujredSi da ase Semdeg (Tan 

vimeorebT `zeda marjvena~, `qveda Sua~ da sxva ujredebs). 

_ es cxrili sul sxvanairi cxrilia. mas ara aqvs raimes 

stirqonebi da raimes svetebi. aseTi cxrili Cven jer ar 

Segvxvedria! esaa kanonzomierebis cxrili. mis ujredebSi 

nakvTebi kanonzomieradaa Caxazuli. aba davakvirdeT pirvel 

striqons, radara nakvTebia masSi? pirvel svetSi? (aRmoCndeba, 

rom igive nakvTebia!). meore striqonSi? meore svetSi? 

(aRmoCndeba, rom yvelgan igive nakvTebia: TiTo-TiTo wiTeli 

samkuTxedi, ruxi marTkuTxedi da Savi xuTkuTxedi, oRond 

sxvadasxva TanmimdevrobiT!). maS ra yofila am cxrilis 

kanonzomiereba? {yvela svetSi da yvela striqonSi unda iyos 

erTi wiTeli samkuTxedi, erTi ruxi marTkuTxedi da erTi 

Savi xuTkuTxedi} diax. maS ra unda exazos cariel ujredSi, 

raTa es kanonzomiereba gagrZeldes? ra aklia mesame striqons? 

{wiTeli samkuTxedi} CaxateT! axla SeamowmeT mesame sveti _ 

kanonzomierad aris Tu ara masSi nakvTebi? {diax}. 

_ Tu iciT, es niSani ra niSania? (masw.-i dafaze dawers did 

kiTxvis niSans ?). axla daakvirdiT ricxTa rigs. romeli 

ricxvi unda eweros kiTxvis niSnis nacvlad? {4} ratom? 

{imitom, rom sul meordeba: 7,2,4, 7,2,4...}. 

59/5/! UbnbTj! `hvebTj! yfmjt! Dbzpgb! eb! hbnpdopcb~/ masw.-i 

naWris gudaSi Cayris plastmasis an sqeli muyaos nakvTebs: 

wreebs, rgolebs, kvadratebs, mkveTrad wagrZelebul marTkuTxedebs, 

samkuTxedebs, tolgverdebian an eqvskuTxedebs an 

rvakuT-xedebs (an erTs, an meores _ radganac isini 

erTmaneTisagan Zne-li gansarCevia!) da ramdenime uswormasworo 

nakvTs. moswavleebma rig-rigobiT unda Cayon gudaSi xeli, 

aiRon erTi nakvTi da gudaSi Cauxedavad, xeliT mosinjviT 

gamoicnon, ra aris (uswormasworoze unda Tqvan _ sxva 

nakvTi). pasuxis Semdeg moswavle amoiRebs nakvTs da yvela 

naxavs, SeeSala Tu ara. Semdeg nakvTs isev Caabruneben gudaSi, 

masw.-i aurevs da momdevno moswavles Caayofinebs xels. 

TamaSis dawyebamde masw.-i klass kargad uCvenebs, Tu radara

- 159 - 

nakvTebs Cayris gudaSi da urCevs, daimaxsovron. 

59/7/! imdagvarive `kanonzomierebis cxrilebia~, rogoricaa 

48.3-Si. masw.-ma es winaswar unda uTxras klass. unda naxon, 

radara nakvTebia pirvel svetSi, pirvel striqonSi, meore 

svetSi, meore striqonSi da ase Semdeg. pirveli cxrilis 

qveda marjvena cariel ujredSi unda iyos Savi wre. meore 

cxrilis cariel ujredSi ki _ amgvarad daxrili monakveTi: /. 

cxadia, amas moswavle Sin unda mixvdes, damoukideblad. 

59/8/! iseTivea, rogoricaa 48.3-is ricxvTa rigi. esec unda 

iTqvas. unda Seivsos marjvniv ori carieli adgili. 

g 49. Ubwj!7/2.jt!hbnfpsfcb/!Tflsfcb.hbnplmfcb!sjdywUb!ljcf{f/ 

5:/1/!48.8 _ SemoqmedebiTia. 

5:/2/! Ubwj! 7/2.jt! Tfkbnfcb! eb! hbnfpsfcb/! _ jer gavimeoroT 

yvela cifri, romlebic viswavleT. sul ramdeni cifri viswavleT? 

{cxra} vin CamoTvlis? kidev romeli maTematikuri niSnebi 

viswavleT? ramdeni? vin CamoTvlis? {metobis, naklebobis, 

tolobis, Sekrebisa da gamoklebis niSnebi, sul xuTi}. 

Semdeg rig-rigobiT waikiTxaven 49.1-is wiTel CarCoSi 

tolobebsa da utolobebs. 

Semdeg Seavseben cariel ujredebs pirveli nimuSis magvarad. 

cota Znelia gamoklebis SemTxvevebi. masw.-i miexmareba: _ 

risi tolia 4_3? saidan ici? rogor unda davawyoT oTxi 

wre, rom es davinaxoT? gamodi dafasTan da daawyve (an: 

daxate)! axla xeli daafare sameuls. ra dagvrCa? ese igi, ra 

unda CavxatoT rveulSi, cariel ujredebSi? 

Semdeg dafaze masw.-i wers: 

4 5 9 _ aq saWiroa Sekreba. sad ra niSnebi CavweroT? 

4 9 _ aq saWiroa Sedareba. sad ra niSani CavweroT? 


9 6 3 _ aq saWiroa gamokleba. sad ra niSnebi CavweroT? 


5:/3/! Tflsfcjtb! eb! hbnplmfcjt! Tftsvmfcb! sjdywUb! ljcjt! 

nfTwfpcjU/!am safexurs Zalian cota dro eyofa.

- 160 - 

keTdeba Sekreba-gamoklebis Cveulebrivi savarjiSoebi jer 

zepirad, Semdeg ki Sedegi Semowmdeba ricxvTa kibeze. 

5:/4/!nsbwbmlvUyfejt!)kfs!bn!ufsnjojt!hbsfTf*!hwfsej/!nbUj! 

ebUwmb/! nsbwbmlvUyfejt! hwfsej! `! npoblwfUjb/! bn! npoblwfUjt! 

cpmpfcj-!cpmp!xfsujmfcj/!es sakiTxi klasma, arsebiTad, ukve 

icis. gavimeorebT, SevajamebT da ganvaviTarebT am codnas. 

_ am suraTze (49.3) marcxniv ra aris daxazuli? 

ramdenkuT-xedia? {xuTkuTxedia} saidan ici? {xuTi gverdi 

aqvs} ramdeni gverdia gamuqebuli wiTlad? {erTi} romeli 

gverdia gamuqebuli _ marjvena Tu marcxena? {marjvena} ra 

nakvTia gverdi? wrea, marTkuTxedia, wertilia, monakveTia, 

kvadratia Tu ra nakvTia? {gverdi _ monakveTia} lurjad 

moniSneT am gverdis, monakveTis boloebi, bolo wertilebi. 

ramdeni bolo aqvs monakveTs? {ori} orive lurjad moniSneT! 

axla Tqvenc gaamuqeT kidev erTi, sxva gverdi, romelic 

gindaT, oRond iseTi, romelsac sxva boloebi eqneba! orive 

bolo wertili sxva unda hqondes! am gamuqebuli gverdis 

orive boloc moniSneT, oRond mwvaned! sul ramdeni gverdi 

aqvs am nakvTs? {xuTi} gverdiT dasviT imdenive wiTeli 

wertili, ramdenicaa gverdebis raodenoba! {wertilTa 

xuTeuli unda daisvas}. 

Semdeg gakeTdeba yvelaferi iseve, oRond marjvena nakvTisaTvis. 

moswavleebi Tavad gaamuqeben rig-rigobiT am nakvTis: 

yvelaze marjvena gverds, yvelaze marcxena gverds, zemoTa 

gverds da qvemoTa gverds. sxvadasxva ferebad moniSnaven am 

gverdebis bolo wertilebs. gverdiT dasvamen imdenive wiTel 

wertils, ramdeni gverdic aqvs nakvTs {rvakuTxeds}. 

5:/5/! `sjdywUb! ebnxlsjwfcb~/ igivea, rac 40.4, oRond 

`dominoebis~ nacvlad moswavleebs urigdeba moZravi cifrebi 

1-dan 9-mde. 

g 50. `Tftblsfcj~/!nsbwbmlvUy/!spnfmjnf!hwfsejt!hbyb{wb/! 

61/1/! 49.9-Si `Camalulia~ yvela naswavli maTematikuri 

niSani (garda cifrebisa): < > = + da _ .!

- 161 - 

61/2/!ufsnjoj!`Tftblsfcj~/!qjswfmj!Tftblsfcj!eb!nfpsf!Tftb. 

lsfcj/!_ amixseniT, ras niSnavs sityva `SesaRebi~ {is, rac unda 

SeiRebos} sityva `Sesalagebeli~? {is, rac unda Selagdes} 

`SesaWmeli~? {is, rac unda SeiWamos} `SesaWreli~? {is, rac 

unda SeiWras} Sesawvavi? {is, rac unda Seiwvas} axla vin mixvdeba, 

ras niSnavs sityva `Sesakrebi~? {is, rac unda Seikribos}. 

masw.-i dafaze dawers: 7 + 1 = 8. _ es niSani ras 

gvibrZanebs? {Sekrebas} risa da ris Sekrebas? {Svidisa da 

erTisa} ese igi, unda Seikribos Svidi da erTi. romeli 

ricxvebi iqneba aq Sesakrebebi? {Svidi da erTi} maTgan 

romelia pirveli Sesakrebi? {Svidi} meore Sesakrebi? {erTi} 

diax, Svids hqvia pirveli Sesakrebi, xolo erTs hqvia meore 

Sesakrebi. ratom? {pirvelad 7 weria, meored _ 1}. 

masw.-i dawers sxva magaliTebs: 6 + 2 = 8, 

2 + 6 = 8, 5 + 3 = 8, 3 + 5 = 8, 2 + 2 = 4. 

_ miCveneT pirveli Sesakrebi! meore Sesakrebi! 

Semdeg rig-rigobiT waikiTxaven msr-Si (50.1) wiTel CarCoSi 

Caweril tolobebs. wakiTxvis Semdeg daevalebaT, rom yvelgan 

pirveli Sesakrebi Semoxazon wiTlad, meore Sesakrebi _ mwvaned. 

61/3/!Ubwj!6/3.jt!hbnfpsfcb!eb!Tfkbnfcb/! 

damuSavdeba sami amocana raodenobis matebaze da sami _ 

klebaze, masw.-i advilad Seadgens maT (amocanaTa nimuSebi da 

maTi damuSavebis wesi ix. 37.2 da 38.2-Si). 

61/4/! nsbwbmlvUyfejt!)kfs!bn!ufsnjojt!hbsfTf*!spnfmjnf! 

hwfsejt! hbyb{wb/! hbnpjlwfUfcb! UwjU! ft! mphjlvsj! ufsnjojd! 

`spnfmjnf~/! keTdeba igive, rac 49.3-Si, oRond gamoikveTeba 

logikuri termini `romelime~. _ gaxazeT romelime gverdi! 

ramdeni gverdi unda gaxazoT? {erTi} romeli gverdi unda 

gaxazoT? {romelime} mainc romelia es romelime gverdi? 

{sul erTia, romelic mogvindeba, is unda gavxazoT}. 

Semdeg boloebs moniSnaven lurjad. 

61/5/!`tbjebobb!ynb@~ moswavle gamodis da dgeba Sua oTaxSi, 

merxebs Soris. masw.-i mas Tvalebs auxvevs. Semdeg masw.-i 

aniSnebs romelime sxva moswavles, romelic adgilidan

- 162 - 

asaxelebs romelime maTematikur sityvas, rac maTematikis 

gakveTilebze aqvT naswavli. Tvalebaxveulma moswavlem unda 

gaimeoreos es sityva da Semdeg Tqvas: visi xma moesma; da rac 

mTavaria _ saidan moesma xma: winidan, marjvnidan, ukana 

marcxena mxridan da ase Semdeg. 

g 51. `bsdfsUj~/!djgsj!1/!Tfls/!dysjmj/!nsbwbmlvUy/!`lvUyf~/ 

62/2/!sbpefopcb!`bsdfsUj~/!sjdywj!ovmj/!djgsj!1/! 

dafaze daxazulia ramdenime elifsi `dominoebiT~ (ise, 

rogorcaa mTw-Si), TviTeul elifss zemodan aweria saTanado 

jamic. bolos xatia erTi carieli elifsi. _ aq ramdeni wrea? 

{cxra} rogoraa dawyobili cxra wre? {eqvseuli da sameuli} 

romelia pirveli Sesakrebi? {eqvsi } meore Sesakrebi? {sami}. 

ase ganvixilavT yvela elifss. mivadgebiT ukanasknel, cariel 

elifss. _ aq ramdeni wrea? {arcerTi} wreebis raodenoba aq 

tolia cxrisa (daadebs TiTs), aq tolia rvisa (daadebs), aq tolia 

eqvsisa (daadebs). aq ki (TiTs daadebs) wreebis raodenoba 

tolia nulisa. ramdeni yofila nuli wre? {arcerTi} diax, nuli 

_ arcerTia. nuli ase iwereba _ 0 (amas zemodan daawers cariel 

elifss). aq ra ricxvi weria (msr-Si 51.1)? {nuli}. 

Semdeg masw.-i dafaze dawers: 


romelia pirveli Sesakrebi? meore Sesakrebi? 



7 6 1 _ aq saWiroa gamokleba. sad ra niSnebi CavweroT? 


romelia pirveli Sesakrebi? meore Sesakrebi? 

----------------------------------------------------------------------------------- 

JJ.7/! Tflsfcjtb! eb! hbnplmfcjt! dysjmfcj/ viwyebT axlebur 

cxrilebs. iseve rogorc ricxvTa kibis SemTxvevaSi, Sekrebagamoklebis 

cxrilebSic ricxvebi jer dominoseburadaa Casmuli, 

meore gakveTilze ki dominoseburadac da cifrebiTac. es ar 

iqneba Zneli. TviT cxrilis Sevseba SeiZleba gauWirdes zog 

moswavles, magram amas sami sagakveTilo monakveTi eTmoba da male

- 163 - 

aRar gauWirdebaT. yvela am safexurs kidev erTi mniSvnelovani 

daniSnuleba aqvs: masw.-ma xSir-xSirad unda ixmaros pirveli xaziT 

naswavli terminebi: Sesakrebi, jami, saklebi, sxvaoba da maklebi 

(cxadia, roca isini naswavli iqneba). TviTonac sul unda ixmaros 

da moswavleebsac aTqmevinos. garda amisa, masw.-ma cxrilisa da 

ricxvTa kibis Temebi zepiri Sekreba-gamoklebis savarjiSoebis 

sakeTeblad unda gamoiyenos. 

----------------------------------------------------------------------------------- 

62/3/!Tflsfcjt!Tfwtfcvmj!dysjmjt!tusjrpoj!eb!twfuj/! 

dafaze momzadebulia iseTive cxrili, rogoricaa msr-Si 

(51.2). mTeli msjeloba mimdinareobs am cxrilis svetebze, 

striqonebsa Tu ujredebze, moswavleebi Tan msr-Si aseTive 

cxrils avseben (`dominosebri ricxvebiT~). 

jer gairkveva, ramdeni sveti, ramdeni striqoni aqvs; ra weria 

zeda marcxena da ase Semdeg ujredebSi; ra weria meore 

svetis pirvel Tu mesame striqonis mesame ujredSi ... ... . 

_ risi striqonia meore striqoni? {erTeulisa} meoTxe? 

{wyvilisa} mesame? {sameulisa} (aseve _ svetebic). ra weria 

sul zeda marcxena ujredSi? {Sekrebis niSani} es niSani risi 

brZanebaa? ras gvibrZanebs, ra gaakeTeTo? {SekribeTo!} ra 

weria meore striqonis meore ujredSi? {oreuli} ratom? 

ratom weria ori wre? {imitom, rom erTeulis striqonSia 

da erTeulis svetSi, erTs mivumatoT erTi ki oris tolia} 

aseve ganixileba danarCeni ujredebic, carielebi ki Seivseba. 

_ es cxrili _ Sekrebis cxrilia. imitom weria aq kuTxeSi 

Sekrebis niSani! 

garda am sakiTxisa, xdeba aqtiuri gameoreba sityvisa àmdenive~, 

oRond ukve sityvier doneze, da ara mxolod nivTebisa 

Tu naxatebis doneze, rogorc iyo aqamde. am sityvis kargi 

codna ramdenime gakveTilis Semdeg yovlad aucilebeli iqneba. 

_ nikas xuTi fanqari aqvs, amdenive aqvs giorgis. ramdeni 

fanqari aqvs giorgis? 

_ klasSi rva gogonaa da amdenivea biWi. ramdeni biWia 

klasSi?

- 164 - 

saWiroa kidev ori amdagvari SekiTxva. 

62/4/! nsbwbmlvUyfejt! )kfs! bn! ufsnjojt! hbsfTf*! `lvUyf~! 

)xwfsp*/!nbUj!ebUwmb/!_ davakvirdeT am oTaxis iataks. romeli 

nakvTia es iataki? {marTkuTxedi} ramdeni kuTxe aqvs mas? 

{oTxi} marTkuTxeds (masw.-i dafaze daxatavs) ramdeni kuTxe 

aqvs? {oTxi} gamovides ia da gviCvenos! 

Semdeg daixazeba sxva ramdenime mravalkuTxedi. am 

safexurze moswavle mxolod TiTis dadebiT aCvenebs, rom ài, 

esaa pirveli kuTxe, meore kuTxe da ase Semdeg.~ meti araferi 

jer araa saWiro. 

bolos, gaarkveven, romeli nakvTebia mocemuli suraTze 

(51.3) da wiTlad moniSnaven maT yvela kuTxes. TviTeuli 

nakvTis gverdiT sicarielea. iq dasvamen wertilebs nakvTis 

kuTxeebis ganlagebis magvarad _ mxolod wertilebs 

(wveroebs). amiT Semzaddeba is sakiTxi, rom `kuTxe~ sinamdvileSi 

wveroa, wertili. Tumca, jer-jerobiT termini `wvero~ 

ar Semogvaqvs. 

TfojTwob; vimeorebT, rom namdvil, geometriul kuTxes 

dawyebiT klasebSi ar vaswavliT. arc gvWirdeba. dagvWirdeba 

mxolod mravalkuTxedis wvero, romelic aris wertili (mezobeli 

gverdebis saerTo wertili). 

62/5/! `txsbgj! qbtvyfcj~! `!`.jU! nfu.oblmfcpcb~/! masw.-i 

asaxelebs ricxvs, moswavlem unda daasaxelos ricxvi: pirvel 

wreze _ erTiT meti, meore wreze _ erTiT naklebi, mesameze _ 

oriT meti, meoTxeze _ oriT naklebi da ase Semdeg. 

èrTiT naklebs~ da òriT mets~ masw.-i winaswar ar 

axsenebs, axsenebs mxolod saTanado wris CamTavrebis Semdeg. 

g 52. 1.jt!Tfebs/!Tflsfcjt!dysjmjt!Tfwtfcb/!hbnfpsfcb/ 

63/2/!ovmjt!Tfebsfcb!obtxbwm!sjdywfcUbo/!!am sagakveTilo 

monakveTs cota dro eyofa. 

_ ra aris raodenoba nuli? {arcerTi} rogor ggoniaT, 

romelia meti _ nuli Tu ori? {ori} saidan ici? {nuli 

arcerTia, ori masze metia} romelia meti _ nuli Tu erTi?

- 165 - 

{erTi} saidan ici? {nuli arcerTia, erTi masze metia} Cven 

rom ricxvebi gvaqvs naswavli, maTgan romelimeze (am sityvis 

gamokveTiT) meti Tu aris nuli? {ara, nuli yvela maTganze 

naklebia} yvela is ricxvi metia, vidre nuli. nuli naklebia, 

vidre TviTeuli es ricxvi! 

Semdeg rig-rigobiT waikiTxaven msr-Si (52.1) wiTel 

CarCoSi Caweril utolobebs. 

63/3/! Tflsfcjt! dysjmjt! Tfwtfcjt! lbopo{pnjfsfcb/! 

eblbwTjsfcb ufsnjoUbo! `Tftblsfcj~/ aq ukve cifrebiani Sekrebis 

cxrilicaa (msr-s 52.2). cxrilis dafaze daxazva aRaraa 

aucilebeli, Tuki moswavleebs gauWirdebaT, maSin daixazos. 

keTdeba igive, rac 51.2-Si, oRond sityvierad kargad 

Camoyalibdeba, Tu ratom weria esa Tu is ricxvi amaTuim 

ujredSi. Semdeg `dominoebiT~ Seivseba meore cxrili, Tan 

kvlav sityvierad Camoyalibdeba, ra Tavsdeba amaTuim ujredSi. 

magaliTad: _ es ujredi meore svetSia da meoTxe striqonSi. 

erTeulis svetSi da sameulis striqonSi. ese igi, masSi unda 

iyos erTs mimatebuli sami cali wre, wreebis oTxeuli. 

emateba erTi sakiTxi: TviTeul SemTxvevaSi masw.-i ekiTxeba: 

_ rasa da ras kreb? romelia pirveli Sesakrebi? meore 

Sesakrebi? es pirveli Sesakrebi sad unda Caiweros? am 

ujredSi romeli Sesakrebi weria? (da ase Semdeg). 

pirveli xaziT naswavl am terminebze sagangebod maxvildeba 

yuradReba. termini `jami~, cxadia, jer araa saWiro. 

garda am sakiTxisa, xdeba xanmokle gameoreba sityvisa 

àmdenive~. SekiTxvebis nimuSebi ix. 51.2-Si (cxadia, masw.-i maT 

sayofacxovrebo Sinaarss Secvlis). 

63/4/!Ubwj!5/4.jt!hbnfpsfcb!eb!Tfkbnfcb/!suraTze (52.3) mocemulia 

nakvTebi. gaarkvioT, TviTeuli ramdenkuTxedia, Tuki 

oTxkuTxedia, xom ar aris kvadrati, marTkuTxedi. _ daakvirdiT 

zeda striqonis nakvTebs. ramdeni nakvTia zeda striqonSi? 

{eqvsi} ramdeni gverdi aqvs gamuqebuli TviTeul maTgans? {er- 

Ti, TiTo gverdi aqvT gamuqebuli} Tqven mwvaned SeaferadeT 

kidev TiTo gverdi, romelic gsurT! Semdeg wiTlad moniSneT

- 166 - 

TiTo kuTxec. axla daakvirdiT qveda striqonis nakvTebs. 

ramdeni nakvTia qveda striqonSi? {imdenive, eqvsi} ra aqvs 

moniSnuli pirvel nakvTs? {kuTxeebi} ramdeni kuTxe aqvs? 

{eqvsi} ramdenkuTxedia es nakvTi? {eqvskuTxedi}. 

Semdeg rig-rigobiT gairkveva, ramdenkuTxedebia danarCeni 

nakvTebi. TviTeul maTgans Savad gaumuqdeba TiTo gverdi da 

wiTlad moeniSneba or-ori kuTxe. 

63/5/! `txsbgj! qbtvyfcj~! `! sbnefoj! UjUjb/ ix. 22.4. oRond 

TiTebis raodenoba aRwevs cxramde, Tanac, Semodis nulic: 

zogjer maswavlebe-li aCvenebs momuWul xels, maSin moswavlem 

unda Tqvas _ nuli. 

g 53. `kbnj~/!Tflsfcjt!dysjmj/ lvcjt!xbyobh/!eb!nbUj!ebUwmb/ 

64/2/!ufsnjoj!`kbnj~/! masw.-s motanili aqvs mogrZo, elifsis 

magvari formis mqone jami (WurWeli). 

dafaze `dominoseburad~ dawyobilia wreebis xuTeuli da 

oTxeuli. _ aq ramdeni wrea? {xuTi} aq? {oTxi} modi, gaga, es 

grovebi Sekribe, oRond Sekribe ai am jamSi! (gaga yvela wres 

moagrovebs da jamSi Cayris. masw.-i amasobaSi dafaze carciT 

daxatavs iseTive xuTeuls, gverdiT iseTive oTxeuls da maT 

erTad Semoavlebs elifss _ ise, rogorcaa mTw-Si da msr-s 

53.1-Si). ai Cven eseni (miuTiTebs daxatulebze) SevkribeT jam- 

Si. ai es (miuTiTebs elifsze) viTom jamis naxatia! romelia 

pirveli Sesakrebi? {xuTi} meore Sesakrebi? {oTxi} jamSi sul 

ramdeni wrea? {cxra} diax, jamSia cxra. xuTi pirveli Sesakrebia, 

oTxi _ meore Sesakrebia. cxras ki davarqvaT jami. 

ramdeni yofila xuTisa da oTxis jami? {cxra} jamia cxra. 

Semdeg daaTvaliereben suraTs (53.1). TviTeuli SemTxvevisaTvis 

rig-rigobiT ityvian: romelia pirveli Sesakrebi, 

meore Sesakrebi, jami. 

aq gavaxsenoT SesakrebTa gadanacvlebadoba. masw.-i dawers 

2+5 da 5+2. _ ras ityviT am jamebis Sesaxeb? {tolia} diax, 

gind ors miumate xuTi da gind xuTs miumate ori, jami igive 

iqneba. Sesakrebebi romc gadavanacvloT (TiTebis trialiT uC-

- 167 - 

venebs `gadanacvlebas~!) jami mainc igive darCeba! jami ar Seicvleba! 

axla Tamarma gviTxras amis sxva magaliTi! jimSermac! 

masw.-i zogad daskvnasac Camoayalibebs, Tumca ara aqvs azri, 

rom es daskvna moswavleebsac gavameorebinoT `wesad~. 

ebtlwob; Sesakrebebi romc gadavanacvloT, jami mainc ar 

Seicvleba. 

64/3/! sjdywjt! npUbwtfcb! Tflsfcjt! dysjmjt! tbUbobep! vk. 

sfeTj/ eblbwTjsfcb! ufsnjoUbo! `kbnj~/! ! dafaze winaswar unda 

iyos momzadebuli iseTive cxrili, rogoricaa msr-Si (53.2). 

mTeli momdevno msjeloba unda warimarTos dafaze daxazul 

cxrilze saTanado svetebis, striqonebisa Tu ujredebis 

miTiTebiT. Tan moswavleebi msr-s cxrils daakvirdebian da 

Seavseben. 

keTdeba igive, rac 52.2-Si: dawyebuli cxrilis svetebisa 

da striqonebis daTvliTa da sxva, damTavrebuli ramdenime 

ujredSi saTanado `dominos~ CaxatviT, oRond winaswari 

sityvieri dasabuTebis moTxovniT. emateba erTi sakiTxi _ 

termini `jami~. 

SekiTxvis nimuSebi: _ jami romel ujredSi unda Caiweros? 

ratom? am ujredSi risi da risi jami unda Caiweros? es 

pirveli Sesakrebi sad unda Caiweros? am ujredSi romeli 

Sesakrebi weria? (da ase Semdeg). 

terminebze `Sesakrebi~ da `jami~ sagangebod maxvildeba 

yuradReba. garda amisa, xdeba xanmokle gameoreba sityvisa 

àmdenive~. SekiTxvebis nimuSebi ix. 51.2-Si (cxadia, masw.-i maT 

sayofacxovrebo Sinaarss Secvlis). 

------------------------------------------------------------------------------------ 

JJJ.6/! nsbwbmlvUyfejt!xwfspfcj/! vixsenebT kubis wveroebsa da 

maT daTvlas. kvadratis `kuTxe~ gaigivdeba kubis wverosTan. 

Semogvaqvs kvadratisa da, sazogadod, mravalkuTxedis wvero 

marTebuli termini, `kuTxis~ nacvlad. mravalkuTxedis gverdebisa 

da wveroebis (kuTxeebis) raodenobaTa cxrili cxrilis axali 

saxea. mravalkuTxedis gverdebisa da kuTxeebis (wveroebis) 

raodenobaTa toloba.

- 168 - 

masw.-isTvis aq mravali terminia naxsenebi: waxnagi, piramida, 

prizma, igiveoba da sxva. arcerTi maTgani man ar unda axsenos 

klasSi. moswavleebs vaswavliT mxolod erT axal termins `wvero~. 

garda amisa, gamovkveTT ukve cnobil termins ìgive~. 

raki geometriul kuTxes dawyebiT klasebSi ar vaswavliT, SeiZleba 

gaCndes SekiTxva araamozneqili mravalkuTxedis kuTxis SemTxvevaSi, 

romlis sididecic 180 gradusze metia. magram saqme isaa, 

rom bavSvi am SemTxvevaSic xedavs mxolod wveros, xolo kuTxes 

kidev ufro naklebad aRiqvams, vidre 180 gradusze nakleb SemTxvevaSi. 

amitom bavSvisTvis araviTari damatebiTi siZnele ar Cndeba. 

erTaderTi, misTvis SeuZlebeli iqneboda zustad 180-gradusiani, 

anu gaSlili kuTxis gaazreba radgan mas ar etyoba wvero. 

kanonzomierebis Tema sagakveTilo monakveTSi aq ukanasknelad 

iqneba. awi igi mxolod saSinao davalebis amocanebSi Segvxvdeba, 

Tumca Zalian xSirad. cxadia, am amocanaTa amoxsnebi klasSi, 

Cveulebriv, gairCeva. amgvarad ganviTardeba kanonzomierebis Tema. 

sasurvelia, kanonzomierebis amocanebis ganxilvisas masw.-ma 

dasvas xolme damatebiTi SekiTxva: 

1) mainc ra meoreba aq mravaljer? 

garda amisa, masw.-ma moswavleebs yuradReba unda miaqcevinos 

imaze, rom amaTuim amocanis kanonzomiereba hgavs sxvas, adre 

ganxiluls. magaliTad, 57.7 hgavs 57.3-s. riT hgavs? 

pirvelklasels albaT gauWirdeba pasuxis zustad Camoyalibeba, 

magram es arcaa saWiro. mTavaria, rom bavSvma daiwyos saTanado 

msjeloba da erTi azri mainc gamoTqvas. 

------------------------------------------------------------------------------------ 

64/4/!lvcjt!xbyobhfcj!)bn!ufsnjojt!hbsfTf"*!`!lwbesbufcj!eb! 

nbUj!ebUwmb/!bn!xbyobhfcjt!`hwfsefcj~!)bov!lvcjt!xjcpfcj!`!bn! 

ufsnjojt! hbsfTf*/! am sagakveTilo monakveTs cota dro eyofa. 

masw.-i jer zepirad, TvalsaCinoebis gareSe dasvams SekiTxvas: _ 

gaixseneT: ra nakvTs davinaxavT, kubs rom winadan SevxedoT? 

{kvadrats} (daanaxvebs da kidev erTxel darwmundebian). 

ramdeni gverdi aqvs kvadrats? {oTxi} gamovides konstantine 

da gviCvenos yvela {erTi, ori, sami, oTxi}.

- 169 - 

kidev davTvliT erTi romelime (am sityvis gamokveTiT) 

waxnagis _ kvadratis gverdebs. Semdeg sxva waxnagisa, Semdeg _ 

kidev sxvisa da davrwmundebiT, rom yovel (am terminis 

gamokveTiT) waxnags hqonia 4 gverdi. _ amasac oTxi gverdi 

aqvs, amasac, amasac. yovel kvadrats xom oTxi gverdi aqvs! 

termini `wvero~ ar Semogvaqvs _ igi Semdeg gakveTilze 

iqneba. 

64/5/ masw.-s xelSi cifrebiani (0-dan 9-mde yvela cifri) 

baraTebis dasta ukavia da rig-rigobiT, sxvadasxva moswavles 

myisierad uCvenebs gverdiT gadabrunebul (`dawvenil~) an 

Tavdayira gadatrialebul romelime cifrs. moswavlem unda 

daasaxelos es cifri. meore wreze _ am cifris momdevno anu 

Semdegi cifri; mesame wreze _ wina cifri da ase Semdeg: wina, 

momdevno, wina, momdevno... 

g 54. sjdywUb!sjhj/!hbnplmfcjt!dysjmj/!lvcjt!xwfspfcj/! 

65/2/!sjdywUb!sjhj;!1-!2-!3-!4-!5-!6-!7-!8-!9-!:/! 

am sagakveTilo monakveTs cota dro eyofa. 

masw.-i cxra moswavles daurigebs TiTo-TiTo moZrav cifrs: 

1, 2, ... , 9. _ gamodiT da swori TanmimdevrobiT daalageT es 

cifrebi! kargia! axla natom (romelsac cifri ar Sexvedria) 

gviTxras, ratom daalages ase? ratom daiwyes erTianiT? mere 

ki oriania. ratom? {jer yvelaze patara ricxvia, mere _ masze 

meti da ase Semdeg, bolosaa cxriani imitom, rom cxra 

yvelaze didia} axla me malxazs vaZlev nulians (aZlevs 

moZrav nulians). gamodi da dade Tavis adgilas am rigSi! 

{dadebs erTianis win} ratom dade aq? {imitom, rom nuli 

yvelaze pataraa}. 

_ axla gadaSaleT rveulebi da wiTel CarCoSi Caweril 

cifrebs daakvirdiT. sworadaa isini dalagebuli? {ki}. 

65/3/!hbnplmfcjt!Tfwtfcvmj!dysjmjt!tusjrpoj!eb!twfuj/! 

yvelaferi gakeTdeba iseve, rogorc 51.2-Si, oRond Sekrebis 

nacvlad iqneba gamokleba. gairCeva cxrili 54.2.! 

65/4/! lvcjt! xwfspfcj/! nbUj! ebUwmb/! wbnlwjesfcU! ufsnjot!

- 170 - 

`xwfsp~/!masw.-i aCvenebs did kubs. iqve devs samkuTxa prizma da 

oTxkuTxa piramida. yvelaze gamokveTili cali wvero piramidas 

aqvs, yvelaze metad igi hgavs namdvil wveros. amitom amiT 

viwyebT. masw.-i TiTs daadebs piramidis zeda wveros da 

ikiTxavs _ ai es ra aris? me mTlian nivTze ki ar gekiTxebiT, 

ara, mxolod ai am adgilze gekiTxebiT! ra aris es? {wveti, 

wveri, wvero} diax, esaa wvero. wveroebi sxvaganacaa, sadac ki 

wvetiviTaa! gamovides naTia da daTvalos amis yvela wvero! 

(daTvlis da aRmoCndeba xuTi wvero). axla zurabi gamovides 

da daTvalos amis (prizmaze miuTiTebs) yvela wvero! (daTvlis 

da aRmoCndeba eqvsi wvero). 

Semdeg masw.-i ramdenime merxze Camoarigebs sxvadasxva zomis 

kubebs, 5-8 cals. _ axla daTvaleT kubis yvela wvero. 

ramdeni wvero aqvs kubs? {rva}. am kubs? amas? {yovel kubs 

rva wvero aqvs}. axla me erT rames vityvi da Tqven miTxariT, 

marTalia Tu ara: rac ufro didia kubi, miT ufro meti 

raodenobis wvero aqvs mas. marTalia? {ara} ratom araa 

marTali? {imitom, rom yovel kubs aqvs rva wvero, sulerTia, 

kubi didia Tu patara!}. rac ufro pataraa kubi, miT ufro 

naklebi raodenobis wvero aqvs mas. baqar (susti moswavle), 

marTalia? {ara} ratom araa marTali? {imitom, rom yovel 

kubs aqvs rva wvero, sulerTia, kubi didia Tu patara!}. 

65/5/!`txsbgj!qbtvyfcj~!`!bohbsjTj/ SekiTxvebia Cveulebrivi 

saangariSo: sams mivumatoT sami; xuTs gamovakloT oTxi da ase 

Semdeg (rac naswavlia). 

g 55. 1.jt!njnbufcb/!hbnplmfcjt!dysjmj/!ufsnjoj!`jhjwf~/ 

66/2/!sjdywjtUwjt!ovmjt!njnbufcb: 5+0 = 5, 9+0 = 9 !eb!tywb/! 

dafaze daxatulia `jamis~ elifsi da Sig rgolebis 

oreuli da eqvseuli (rogorcaa mTw-Si). gverdiT, marjvniv, 

kidev erTi amgvari elifsia da Sig marcxena mxares xatia 

Svideuli, marjvena mxare ki carielia. jer ganvixilavT 

pirvel jams: _ romelia aq Sesakrebebi? {ori da eqvsi} jami 

risi tolia? {rvisa} (masw.-i elifss zemodan daawers: 2 + 6 =

- 171 - 

8). axla vnaxoT aq, marjvniv. risi tolia pirveli Sesakrebi? 

{Svidisa} meore Sesakrebi? {ar gvaqvs, araa, ese igi, nulia} 

jamSi sul ramdenia rgoli? {Svidi} jami risi tolia? 

{Svidisa} diax, Svids mivumatoT nuli udris Svids (elifss 

zemodan daawers: 7 + 0 = 7). Svids rom araferi mivumatoT, 

isev Svidi iqneba! axla vin gvetyvis, risi tolia xuTs 

mivumatoT nuli? {xuTisa} ratom? {xuTs rom araferi 

mivumatoT, isev xuTi iqneba}. 

aseve gairkveva, risi tolia 3+0, 2+0 da sxva ramdenime. 

Semdeg moswavleebi rig-rigobiT waikiTxaven msr-Si wiTel 

CarCoSi (55.1) Caweril tolobebsa da utolobebs. Tan 

Sesabamisad wertilebiT Seavseben Sesabamis ujredebs, romlebic 

qvemoTaa mixazuli. 



ebtlwob; raime ricxvs rom nuli mivumatoT, isev im ricxvs 

miviRebT. 

66/3/!hbnplmfcjt!dysjmjt!Tfwtfcjt!lbopo{pnjfsfcb/! 

jer swrafad gavimeorebT Sekrebis cxrilis Sevsebis kanon- 

zomierebas (52.2). 

Semdeg moswavleebs SekiTxvebis meSveobiT gavaxsenebinebT, 

rom Sekrebis cxrilis pirvel striqonSi Cawerilia pirveli 

Sesakrebebi, xolo pirvel svetSi _ meore Sesakrebebi. svetisa 

da striqonis gadakveTis ujredebSi ki ra iwereba? {jami, risi 

tolicaa jami}. 


Sekrebis nacvlad iqneba gamokleba. viyenebT cxrils 55.2. 

66/4/! lwbesbujt! `lvUyjtb~! eb! lvcjt! xbyobhjt! xwfspt! 

jhjwfpcb/!ufsnjoj!`jhjwf~-!`jhjwfb~/!lwbesbujt!xwfsp/!! 

masw.-i klass aCvenebs did kubs da daaTvlevinebs misi wina 

waxnagis wveroebs. _ ra nakvTia es waxnagi? {kvadratia} gvanca, 

gviCvene am kvadratis kuTxeebi! ramdeni kuTxe aqvs? {oTxi} 

axla am kvadratis wveroebi gviCvene! diax, kvadratis kuTxe da 

wvero igive yofila! igive (am sityvis gamokveTiT). wvero

- 172 - 

igivea, rac kuTxe. ramdeni wvero aqvs kvadrats? {oTxi} axla 

dafaze davxazoT ori kvadrati (daxazavs erTs dauxrelad, 

meores _ cerad, orives Seaferadebs). giorgim gviCvenos am 

kvadratis wveroebi! xatiam ki gviCvenos am kvadratis wveroebi! 

ramdeni wvero aqvs? {oTxi} am kvadratsac oTxi wvero aqvs, 

amasac, amasac. yovel (am sityvis gamokveTiT) kvadrats oTxi 

wvero aqvs! 

Semdeg masw.-i klass aCvenebs did kubTan erTad patara 

kubsac. _ romels aqvs wveroebis meti raodenoba? marTalia 

Tu ara: rac ufro didia kubi, miT ufro meti wvero aqvs 

mas? ratom? 

66/5/! `txsbgj! qbtvyfcj! `! lblvo.ubTfcj~/ ix.18.4, oRond 

raodenoba aRwevs cxramde. 

g 56. 1.jt!!njnbufcb/!hbnplm/!dysjmj/!nsbwbmlvUy/!xwfsp/ 

67/1/! 55.8-is zeda rigSi unda gamoikveTos kanonzomiereba: 

yvelaferi saWmelia, garda erTisa _ Coxisa. swored igi 

arRvevs kanonzomierebas. meore rigSi yvelaferi saaxalwlo 

nivTebia, garda Tovlis papisa, romelic cocxalia, adamiania. 

qveda rigSi yvelaferi imisaTvisaa, rom sadme mivideT (termini 

`transporti~ araa saWiro!), garda oda saxlisa. 

67/2/! sjdywjt! ! ovmjtUwjt! ! njnbufcb: 0+6 = 6, 0+1 = 1!!eb! 

tywb/ 0+0 = 0/! jer swrafad gavameorebinebT 55.1-is 

sakiTxs am saxiT: _ pirveli Sesakrebia cxra, meore Sesakrebi 

_ nuli. risi tolia jami? 

aseve gamoiTvleba kidev ramdenime amdagvarive jami: 4+0, 

0+0... 

Semdeg SekiTxva garTuldeba: _ pirveli Sesakrebia Svidi, 

jamic Svidis tolia. risi tolia meore Sesakrebi? {nulis} 

ratom? {Svids mivumatoT nuli udris Svids _ sxvagvari axsna 

araa saWiro}. 

Semdeg yvelaferi TiTqmis iseve gakeTdeba, rogorc 55.1-Si, 

oRond nuli iqneba ara pirveli, aramed meore Sesakrebi: 

marjvena elifsSi sicariele iyos ara marcxniv (pirveli

- 173 - 

Sesakrebis adgilas), aramed _ marjvniv. 

dafaze daxatulia `jamis~ elifsi da Sig rgolebis sameuli 

da xuTeuli (rogorcaa mTw-Si). gverdiT, marjvniv, kidev 

erTi amgvari elifsia da Sig marjvena mxares xatia oTxeuli, 

marcxena mxare ki carielia. jer ganvixilavT pirvel jams: _ 

romelia aq Sesakrebebi? {sami da xuTi} jami risi tolia? 

{rvisa} (masw.-i elifss zemodan daawers: 3 + 5 = 8). axla 

vnaxoT aq, marjvniv. risi tolia pirveli Sesakrebi? {nulisa} 

meore Sesakrebi? {oTxisa} jamSi sul ramdenia rgoli? {oTxi} 

jami risi tolia? {oTxisa} diax, nuls mivumatoT oTxi 

udris oTxs (elifss zemodan daawers: 0 + 4 = 4). 

aq isev gavaxsenoT SesakrebTa gadanacvlebadoba. masw.-i 

dawers 2+4 da 4+2. _ ras ityviT am jamebis Sesaxeb? 

{tolia} ratom? {Sesakrebebi romc gadavanacvloT, jami mainc 

ar Seicvleba}. 

masw.-i dawers 0+4 da 4+0. _ am jamebis Sesaxeb ras 

ityviT? {tolia} 4+0 ris tolia? {oTxisa} ese igi, 0+4 ris 

toli iqneba? {igivea, oTxisa}. axla vin gvetyvis, risi tolia 

nuls mivumatoT xuTi? nuli? {xuTisa} ratom? 

dasabuTeba orgvari SeiZleba iyos: 1) nuls rom xuTi 

mivumatoT, igivea, rac xuTs nuli mivumatoT, es ki xuTis 

tolia; 2) arafers rom xuTi mivumatoT, xuTi iqneba. 

aseve gairkveva, risi tolia 3+0, 2+0 da sxva ramdenime. 

Semdeg sxvadasxva moswavleebi rig-rigobiT waikiTxaven 

msr-Si wiTel CarCoSi (56.1) Caweril tolobebsa da 

utolobebs. Tan Sesabamisad wertilebiT Seavseben Sesabamis 

ujredebs, romlebic qvemoTaa mixazuli. 



ebtlwob; nuls rom raime ricxvi mivumatoT, isev im ricxvs 

miviRebT. 

67/3/! sjdywjt! npUbwtfcb! hbnplmfcjt! dysjmjt! tbUbobep! 

vksfeTj/! ! ! jer swrafad gavimeorebT ricxvis moTavsebas 

Sekrebis cxrilis ujredebSi (53.2).

- 174 - 


Sekrebis nacvlad iqneba gamokleba. Seivseba cxrili 56.2 

(`dominoebiT~). 

67/4/! tb{phbepe! nsbwbmlvUyfejt! )kfs! bn! ufsnjojt! hbsfTf*! 

xwfsp/! nbsUlvUyfejt-! tbnlvUyfejtb! eb! tywb! obtxbwmj! nsbwbm. 

lvUyfefcjt! xwfspfcjt! ebUwmb/! ! xwfsp! `! xfsujmjb/!! 

daaTvaliereT suraTi (56.3). ramdeni nakvTia daxazuli? {cxra} 

pirveli nakvTi ra nakvTia? {kvadrati} ramdeni wevro aqvs mas? 

{oTxi} moniSneT wiTlad oTxive! meore ra nakvTia? 

{marTkuTxedi} mas ramdeni wvero aqvs? {oTxi} oTxive wiTlad 

moniSneT! (da a. S. bolomde). 

Semdeg zepirad, TvalsaCinoebis gareSe: _ rvakuTxeds 

ramdeni wvero aqvs? {rva} xuTkuTxeds? {xuTi} wvero ra aris: 

monakveTi, xazi, wertili Tu wre? {wertili} diax, nakvTis 

wvero _ wertilia! wertili xom yvela nakvTze ufro pataraa! 

aris raime nakvTi, wertilze patara rom iyos? {ara} yvela 

sxva nakvTi ufro didia, vidre wertili! wertili yvelaferze 

ufro pataraa! amitom wveroc yvelaferze ufro pataraa, 

wvero xom wertilia! 

Semdeg masw.-i klass aCvenebs did kvafrats da patara 

kvadrats. _ romels aqvs wveroebis meti raodenoba? 

marTalia Tu ara: rac ufro didia kvadrati, miT ufro meti 

wvero aqvs mas? ratom? 

67/5/! `txsbgj! qbtvyfcj~! `! ÙjUfcjt! Dwfofcb~/ masw.-i 

moswavles usaxelebs ricxvs: 0, 1, 2, 3, ... , 9. moswavlem unda 

aCvenos amdeni TiTi, oRond aucileblad orive xelis TiTebi 

_ ise, rom erTad iyos dasaxelebuli ricxvi. masw.-i ukve 

ityvis: _ TiTebis raodenobaTa jami (am sityvis gamokveTiT) 

unda iyos is ricxvi, romelsac me getyviT. 

Tuki masw.-i ityvis èrTi~, moswavle aCvenebs erTi xelis 

erT TiTs da meore xelis momuWul muSts (nuli TiTi). Tuki 

masw.-i ityvis `nuli~, moswavle aCvenebs orive xelis momuWul 

muStebs (nul-nuli TiTi).

- 175 - 

g 57. hbnfpsfcb/!lbopo{pnjfsfcjt!ebsSwfwb!bo!Tfwtfcb/! 


gameordeba nuli, Sesakrebebi da jami. amisaTvis daismis 

zepiri SekiTxvebi, romlebsac sxvadasxva moswavleebi upasuxeben, 

SeZlebisamebr swrafad. visac gauWirdeba, gamoiyenos mTw. 

isedac, mTw yvelas gaSlili hqondes. SekiTxvebi: 

1) pirveli Sesakrebia ori, meore oTxi. ras udris jami? 

2) Sesakrebebia sami da eqvsi. ras udris jami? 

3) pirveli Sesakrebia nuli, meore xuTi. ras udris jami? 

4) Sesakrebebia sami da nuli. ras udris jami? 

5) pirveli Sesakrebia ori, jamic oria. ras udris meore 

Sesakrebi? 

6) meore Sesakrebia rva, jamic rvaa. ras udris pirveli 

Sesakrebi? 

bolos, gairCeva SedarebiT Zneli, saazrovno SekiTxvebi: 

7) SeiZleba Tu ara, rom pirveli Sesakrebi ufro meti iyos, vidre 

meore Sesakrebi? {SeiZleba} magaliTad? {xuTs mivumatoT sami}. 

8) SeiZleba Tu ara, rom meore Sesakrebi ufro meti iyos, vidre 

pirveli Sesakrebi? 

9) SeiZleba Tu ara, rom meore Sesakrebi ufro naklebi iyos, 

vidre pirveli Sesakrebi? 

10) SeiZleba Tu ara, pirveli Sesakrebi ufro naklebi iyos, 

vidre meore Sesakrebi? 

11) SeiZleba Tu ara, rom meore Sesakrebi pirveli Sesakrebis 

toli iyos? 

12) SeiZleba Tu ara, rom Sesakrebebi erTmaneTis toli iyos? 

13) SeiZleba Tu ara, rom pirveli Sesakrebi jamis toli iyos? 

{SeiZleba, roca meore Sesakrebia nuli}. 

14) SeiZleba Tu ara, rom meore Sesakrebi jamis toli iyos? 

{SeiZleba, roca pirveli Sesakrebia nuli}. 

15) SeiZleba Tu ara, rom pirveli Sesakrebi jamze meti iyos? 

{ar SeiZleba}. 

16) SeiZleba Tu ara, rom jami meore Sesakrebze naklebi iyos? 

{ar SeiZleba}.

- 176 - 

17) SeiZleba Tu ara, rom jami Sesakrebze meti iyos? {SeiZleba}. 

68/3/! Ubwj! 8/3.jt! hbnfpsfcb! eb! Tfkbnfcb/! ! orive cxrili 

(56.7+57.2) Sevsebulia saSinao davalebaSi. Semowmdeba Sevsebis 

marTebuloba, Secdomebi kargad gairkveva. Semdeg gameordeba is, 

Tu Sekrebis cxrilSi sad iwereba pirveli Sesakrebi, sad _ 

meore Sesakrebi, sad _ jami. amisaTvis masw.-i dafaze dawers 

Sekrebisa da gamoklebis TiTo magaliTs. _ romelia aq 

pirveli Sesakrebi? Sekrebis cxrilSi sad unda CavweroT igi? 

romelia aq meore Sesakrebi? cxrilSi sad unda CavweroT igi? 

risi tolia aq jami? cxrilSi sad unda CavweroT igi? 

bolos, sxvadasxva moswavleni rig-rigobiT waikiTxaven 

56.8-is tolobebs. Tan xmamaRla ityvian, romelia pirveli 

Sesakrebi, romeli _ meore Sesakrebi, romelia jami. 

68/4/! bewjmj! bnpdbofcj! mphjlvsj! lbopo{pnjfsfcjt! ebsSwf. 

wb{f!`!spnfmj!bsSwfwt!lbopo{pnjfsfcbt

- 177 - 

68/5/! `txsbgj! qbtvyfcj~/ SekiTxvebi imdagvarivea, rogorebicaa 

57.1-is pirveli eqvsi SekiTxva. maT TamaSis saxe miecema. 

g 58. `tblmfcj~/!Tftb[mfcmpcboj/!hwfse/!eb!xwfs/!dysjmj/! 

68/8/! `! ra meordeba am rigSi ucvlelad? {samkuTxedi} ra 

aris, rom meordeba da Tan rom icvleba? {monakveTi} rogor 

icvleba? {matulobs raodenoba: jer erTia, mere _ ori, sami...}! 

_ wina gakveTilze rom Segvxvda, maTgan romel kanonzomierebas 

hgavs am rigis kanonzomiereba? {57.3} riT hgavs? {iq wre 

ucvlelad meordeboda, patara Savi wreebi ki jer erTi iyo, 

mere _ ori, mere _ sami da ase Semdeg}. 

sasurvelia, kanonzomierebis amocanebSi amgvari 

msjeloba SemdgomSic Catardes xolme. 

69/2/!ufsnjoj!`tblmfcj~/!dafazea `dominoseburi~ Svideulebi 

rgolebisa, erT rigSi ganlagebuli, eqvsi cali. 

_ 7-s gamovakloT 2. 7-s gamovakloT 5. 7-s gamovakloT 4. 

7-s gamovakloT 3. 7-s gamovakloT 6. 7-s gamovakloT 1. 

(gamodis moswavle da waSlis saTanado raodenobis rgolebs). 

_ romeli ricxvidan vaklebdiT sxvadasxva ricxvebs? {Svididan} 

ricxvs, romlidanac vaklebT, saklebi hqvia. saklebi. 

_ amixseniT, ra sWirdeba saferadebels? {gaferadeba} 

sarecxs? {garecxva} saTxrels? {gaTxra} sarbens? {sirbili} 

saklebs raRa sWirdeba? {gamokleba}. 

dafaze msxvilad daiwereba: 9 _ 5 = 4. _ romelia aq 

saklebi? {cxra} diax, romelsac vaklebT (TiTs daadebs 

cxrians), isaa saklebi. rasac vaklebT is ki ara! xuTi (TiTs 

daadebs xuTians) aq araa saklebi! saklebia cxra. Sesakrebebi 

rogor gvqonda? pirveli Sesakrebi da meore Sesakrebi. saklebi 

ki ase ar SeiZleba. pirveli saklebi da meore saklebi _ 

uazrobaa, ar SeiZleba! saklebi mxolod erTia! 


wiTel CarCoSi (58.1) Caweril tolobebs. saklebebs lurjad 

Semoxazaven. qveda ujredebs pirveli ori nimuSis magvarad 

Seavseben.

- 178 - 

bolos, waikiTxaven wiTeli CarCos qvemoT daweril tolobebsa 

da utolobebs. saklebebs kvlav lurjad Semoxazaven. 

----------------------------------------------------------------------------------- 

JJ.8/! jxzfcb! bybmj! Ufnb;! zwfmb! Tftb[mfcmpcjt! bnpxvswb! 

)lpncjobupsjljt! tbxzjtfcj*/ terminebs `Tanmimdevroba~ da 

`SesaZlebloba~ ver gaveqceviT, masw.-i maT Tavidan ixmars 

yovelgvari axsna-ganmartebis gareSe, saWiroebis mixedviT, magram 

yovelTvis konkretul da TvalsaCino viTarebaSi. moswavleebi am 

sityvebis mniSvnelobas mainc ver Caswvdebian, pirvel klasSi es 

arcaa saWiro. ubralod, maT unda icodnen, ra moeTxovebaT, roca 

amocanis piroba moiTxovs: àr gamogrCeT arcerTi Tanmimdevroba 

(SesaZlebloba)!~ amas ki sakmaod male aiTviseben. 

amdagvarive viTareba iyo agreTve terminebTan `kanonzomiereba~ 

(àrRvevs kanonzomierebas~) da `dajgufeba~. isinic mxolod 

konkretuli amocanebis konteqstSi gvxvdeba da amis mets arc 

arafers moveliT maTgan. 

kombinatoruli amocanebis amoxsnisas unda gvaxsovdes zms $ 

7-Si mocemuli axsna-ganmartebani. 

----------------------------------------------------------------------------------- 

69/3/! cbwTwfcjt!bo!ojwUfcjt! hbombhfcb!zwfmb! Tftb[mfcmpcjt! 

bnpxvswjU-!spdb!vdopc!Tftb[mfcmpcbUb!sbpefopcb!bs!bSfnb. 

ufcb! 5.t/! gamoviyvanT sam moswavles. unda dadgnen erTmaneTis 

ukan. sxvadasxvagvarad. magram aseTi wesia: ganlagebis gameoreba 

ar SeiZleba. scadon. masw.-i miaqcevs yuradRebas, rom ar 

gamorCeT arcerTi SesaZlebloba: ia, lia, gia; ia, gia, lia; 

gia, ia, lia; gia, lia, ia; lia, gia, ia; lia, ia, gia. 

SesaZleblobaTa Cawera da daTvla jer araa saWiro. 

_ axla sxvadasxvagvarad davawyoT nakvTebi: samkuTxedi, 

kvadrati da wre. 

gamova erTi Zlieri moswavle da daiwyebs masw.-is 

daxmarebiT, Semdeg TviTon ganagrZobs (magnitur an lursmnebian 

dafaze). samkuTxedebi, wreebi da kvadratebi sakmaod unda 

iyos, raTa moswavles ar mouwios dawyobili ganlagebis 

daSla. ese igi, eqvsive SesaZlo ganlageba bolos erTad iqneba

- 179 - 

dafaze. sxva moswavle sityvierad aRwers: wre, samkuTxedi, 

kvadrati; wre, kvadrati, samkuTxedi; kvadrati, wre, 

samkuTxedi; kvadrati, samkuTxedi, wre; samkuTxedi, kvadrati, 

wre; samkuTxedi, wre, kvadrati. 

bolos, Sesruldeba msr-Si 58.2. qveda cariel CarCoSi unda 

Caxaton is erTi SesaZlo ganlageba, romelic aklia, romelic 

araa. 58.2-is boloSi, uabzacod. 

_ vin da vin midian am suraTebze? {bebo, papa da cuga}. 

jer pirvel suraTs daakvirdiT. pirvelia zeda marcxena 

suraTi. yvelam ipova zeda marcxena suraTi? ra TanmimdevrobiT 

midian? yvelaze win vin midis, mere vin? {cuga, papa, bebo}. 

axla meore suraTi naxeT, zeda Sua suraTi. ra TanmimdevrobiT 

midian? yvelaze win vin midis, mere vin? {cuga, bebo, papa}. aq 

orivegan vin midis pirveli? {cuga}. kargi, axla vnaxoT mesame 

suraTi, zeda marjvena suraTi. ra TanmimdevrobiT midian? 

yvelaze win vin midis, mere vin? {bebo, cuga, papa}. axla 

meoTxe suraTi naxeT, qveda marcxena suraTi. ra 

TanmimdevrobiT midian? yvelaze win vin midis, mere vin? {bebo, 

papa, cuga}. aq orivegan vin midis pirveli? {bebo}. kargi, axla 

vnaxoT mexuTe suraTi, qveda Sua suraTi. ra TanmimdevrobiT 

midian? {papa, bebo, cuga}. dagvrCa kidev erTi suraTi, 

romelic araa Sevsebuli da Cven unda SevavsoT. ar unda 

gavimeoroT iseTi Tanmimdevroba, romelic ukve iyo! erTi 

Tanmimdevroba gvaklia da swored igi unda CavxatoT. romeli 

gvaklia? pirvelad, yvelaze win vin unda iyos? daxedeT, zemoT 

jer vin iyo pirvelad? {cuga} mere? {isev cuga} mere? {bebo} 

mere? {isev bebo} mere? {papa} axla vin unda iyos pirvelad? 

{papa} diax, yvelaze win papa unda midiodes! CaxateT papa 

yvelaze win, ai, ase (masw.-i dafaze daxatavs sqematur kacunas 

wveriTa da yavarjniT, rogorcaa msr-Si}. kargi, meore vin unda 

iyos? bebo? maSin mesame vin iqneba? {cuga} magram daakvirdiT 

mexuTe suraTs _ aq vinaa meore da mesame? {bebo da cuga} me 

xom giTxariT, Tanmimdevroba ar unda gavimeoroT-meTqi! maS, 

pirvelia papa. meore vin unda iyos? {cuga}. mesame? {bebo}. aba,

- 180 - 

SevamowmoT, sadme xom ar arian ase: papa, cuga, bebo? {ara, ase 

arsad ar arian} hoda ese igi, swored es Tanmimdevroba 

gvklebia. CavxatoT! papas mere vin unda iyos? {cuga} mere? 

{bebo} CaxateT! cuga ase daxateT, bebo _ ase (dafaze aCvenebs 

gaadvilebul sqematur nimuSebs}. 

69/4/!lpolsfuvm!nsbwbmlvUyfeUb!hwfsefcjtb!eb!xwfspfcjt! 

sbpefopcbUb!dysjmj/!bn!dysjmjt!tusjrpoj!eb!twfuj/! 

am sagakveTilo monakveTs cota dro eyofa. 

jer klass gavameorebinebT, ra aris wvero da gverdi. 

amisaTvis dafaze daixazeba didi uswormasworo rvakuTxedi da 

masze aCveneben. Semdeg Seivseba cxrili (58.3). 

_ ramdeni striqonia am cxrilSi? {sami} ra aris Caxazuli 

pirveli striqonis ujredebSi? {nakvTebi} meore striqoni ki 

am nakvTebis gverdebis raodenobis striqonia. ramdeni gverdic 

aqvs TviTeul nakvTs. mesame striqoni ki am nakvTebis 

wveroebis raodenobis striqonia. ramdeni wveroc aqvs 

TviTeul nakvTs. moZebneT meore striqonis meore ujredi. 

meore striqonis meore ujredi. romel svetSia es ujredi? 

{kvadratisa} ra aris am ujredSi? {oTxeuli} ratomaa 

oTxeuli? {imitom, rom es sveti _ kvadratis svetia da 

kvadrats oTxi gverdi aqvs} axla vnaxoT mesame striqonis 

meore ujredi. mesame striqonis meore ujredi. ra aris Sig? 

{oTxeuli} ratom? {imitom, rom kvadrats oTxi wvero aqvs}. 

aseve gairCeva mesame, samkuTxedis sveti. Semdeg Seivseba 

meoTxe, oTxkuTxedis sveti. Tan yvelaferi sityvieri 

msjelobiT Camoyalibdeba. 

_ ramdeni svetia Sevsebuli? {sami} ramdeni dagvrCa 

Sesavsebi? {Svidi} es Svidi sveti Sin SeavseT! 

g 59. 1.jt!hbnplmfcb/!Tftb[mfcmpcboj/!dysjmfcjt!Tfwtfcb/! 

6:/2/!sjdywjtUwjt!ovmjt!hbnplmfcb;!:`1 = :-!5`1 = 5!!eb!tywb/! 

jer klass gavameorebinebT ricxvisaTvis nulis mimatebas _ 

ramdenime magaliTs. _ ratom ar icvleba ricxvi misTvis nulis 

mimatebiT? {imitom, rom nuli araferia, arafris mimateba

- 181 - 

ki raodenobas ar Secvlis} axla warmovidginoT ricxvisaTvis 

nulis gamokleba. magaliTad, saklebia 4, mas unda gamovakloT 

nuli. ras miviRebT? {isev oTxs} ratom? {imitom, rom nuli 

araferia, arafris gamokleba ki raodenobas ar Secvlis}. 


wiTel CarCoSi (59.1) Caweril tolobebs. Tan saklebebs 

wiTlad Semoxazaven. 

6:/3/!obybufcjt!hbombhfcb!zwfmb!Tftb[mfcmpcjt!bnpxvswjU-! 

spdb! vdopc! Tftb[mfcmpcbUb! sbpefopcb! bs! bSfnbufcb! 5.t/!! 

ÙjUp.UjUp~/!!_ warmoidgineT, rom gvaqvs tomara, romelSic 

awyvia samkuTxedebi, kvadratebi da wreebi (suraTi 59.2). unda 

amoviRoT TiTo-TiTo (am sityvis gamokveTiT) samkuTxedi, kvadrati 

da wre. ese igi, ra unda amoviRoT? {TiTo samkuTxedi, 

TiTo kvadrati da TiTo wre, unda amoviRoT erTi samkuTxedi, 

erTi kvadrati da erTi wre} isini unda CavxatoT am CarCoebSi 

sxvadasxva TanmimdevrobiT. aq rveulSi ukve Caxazuli ramdeni 

Tanmimdevrobaa? {sami} danarCenebi Cven unda CavxatoT. Tanmimdevroba 

ar unda gavimeoroT! aba, CavxatoT! ar gamogvrCes arcerTi 

Tanmimdevroba. 

gameordeba imdagvarive msjeloba, rogoric iyo 58.2-is 

boloSi, oRond, cxadia, cuga, bebo da papa Seicvleba wriT, 

kvadratiTa da samkuTxediT. qveda CarCoebSi unda Caixatos: 

wre, kvadrati, samkuTxedi; kvadrati, wre, samkuTxedi; 

kvadrati, samkuTxedi, wre. 

6:/4/!lpolsfuvm!nsbwbmlvUyfeUb!hwfsefcjtb!eb!xwfspfcjt! 

sbpefopcbUb! dysjmj/! bnhwbsj! dysjmfcjt! Tfwtfcb/! lpolsfuvm! 

nsbwbmlvUyfeUb! hwfsefcjtb! eb! xwfspfcjt! sbpefopcbUb! 

upmpcb/!iseve, rogorc 58.3-Si, ganixileba da Semdeg Seivseba 

cxrilis (59.3) ori-sami sveti (danarCeni _ iseve saSinao 

davalebaSi). ukanasknel svetSi TviTon unda Caxazon raime 

axali nakvTi! 

Semdeg masw.-i klass aCvenebs did marTkuTxeds da patara 

marTkuTxeds. _ romels aqvs wveroebis meti raodenoba? 

marTalia Tu ara: rac ufro pataraa marTkuTxedi, miT ufro

- 182 - 

naklebi wvero aqvs mas? ratom? 

erTi siaxlea: saboloo daskvna sityvierad Camoyalibdeba 

da gaiazreba: 

ebtlwob; nakvTs yovelTvis imdeni wvero aqvs, ramdenic _ 

gverdi. gverdebi imdenivea, ramdenic _ wveroebi. gverdebisa da 

wveroebis raodenoba tolia. 

6:/5/!Ùwbmjt!ebybnybnfcb~/ ix. 15.4. 

g 60. `tywbpcb~/!Tftb[mfcmpcbUb!bnpxvswb/!hbnfpsfcb/ 

71/2/! ufsnjoj! `tywbpcb~/! dafaze rgolebis `dominoseburi~ 

rvaeulia. _ ramdenia? {rva} dodom gamoaklos amas xuTi 

(gamova da waSlis xuT rgols). ra iyo saklebi? {rva} ra 

dagvrCa? {sami} gamoklebis Semdeg rac dagvrCeba, im ricxvs 

sxvaoba hqvia. sxvaoba. maS, ra aris sxvaoba? aq ra miviReT 

sxvaoba? {sami}. 


saklebi? {Svidi} diax, romelsac vaklebT (TiTs daadebs 

Svidians), isaa saklebi. sxvaoba romelia? {oTxi} diax, rasac 

miviRebT (TiTs daadebs oTxians), isaa sxvaoba. 


msr-Si Caweril tolobebsa da utolobebs (60.1). saklebebi 

ukve yviTladaa Semoxazuli, Sesakrebebi _ mwvaned. sxvaobebs 

Semoxazaven wiTlad. cxadia, sxvaoba mxolod tolobebSia, 

utolobebSi ki sxvaoba araa! 

71/3/! obyb{jt! hbgfsbefcb! zwfmb! Tftb[mfcmpcjt! bnpxvswjU-! 

spdb!vdopc!Tftb[mfcmpcbUb!sbpefopcb!bs!bSfnbufcb!5.t/! 

masw.-i dafaze daxazavs 60.2-is erT wres. _ ramdeni aseTi 

wrea naxazze? (suraTi 60.2) {eqvsi}. es wre unda gavaferadoT 

sam ferad. ra feris fanqrebi gvaqvs? {Savi, wiTeli, mwvane} 

wris erTi nawili (TiTs daadebs) erT ferad unda 

gavaferadoT, meore nawili (TiTs daadebs) _ meore ferad, 

mesame (TiTs daadebs) _ mesame ferad. yvela gaferadeba 

sxvadasxva unda iyos. gaferadebis gameoreba ar SeiZleba! aba, 

gavaferadoT. ar gamogvrCes arcerTi SesaZlebloba! pirveli

- 183 - 

gaferadeba sul erTia, rogori iqneba. gviTxras anam (yvelaze 

pasiuri moswavle) _ rogor gavaferadoT pirveli wre? gamodi 

da gviTxari! {es nawili _ wiTlad, es _ Savad, es _ mwvaned} 

kargi, yvelam ase gaaferados. Senc dajeqi da gaaferade. 

_ rogor gavaferadoT meore wre? modi, gadavanacvloT 

Savi da mwvane, wiTeli ki adgilze davtovoT. ese igi, sadac 

Savia, iq mwvane iyos, sadac mwvanea _ iq Savi iyos. 

_ rogor gavaferadoT mesame wre? modi, axla 

gadavanacvloT wiTeli da mwvane, Savi ki adgilze davtovoT. 

ese igi, sadac wiTelia, iq mwvane iyos, sadac mwvanea _ iq 

wiTeli iyos. 

da ase Semdeg _ umjobesia, TviTon moswavleebma ganagrZon. 

masw.-i ikiTxavs xolme: _ Salvam SemogvTavaza ase: Savi, mwvane, 

wiTeli _ aba daukvirdiT, aseTi gaferadeba ukve xom ar 

gvaqvs? {ki, ukve gvaqvs!} ho, ese igi, es ar varga. aba rogor 

gavaferadoT? (da ase Semdeg). 

SesaZlo gaferadebani: wiTeli, Savi, mwvane; wiTeli, 

mwvane, Savi; mwvane, wiTeli, Savi; mwvane, Savi, wiTeli; Savi, 

mwvane, wiTeli; Savi, wiTeli, mwvane. 

71/4/!Ubwj!6/4.jt!Tfkbnfcb!eb!hbnfpsfcb/!iseve rogorc 59.3- 

Si, Seivseba cxrilis (60.3) ramdenime sveti, sityvieri 

msjelobis TanxlebiT, oRond yvela nakvTs Tavad moswavleebi 

Caxazaven. danarCeni svetebi _ saSinao davalebad miecemaT. 

saWiroa erTi mniSvnelovani sakiTxis dasma da garkveva: _ 

ramdeni striqonia am cxrilSi? {ori} risi striqonia meore? 

{nakvTis gverdebis an wveroebis raodenobisa} ratomaa am 

cxrilSi mxolod ori striqoni? {imitom, gverdebis 

raodenoba igivea, rac wveroebisa; es raodenobani tolia da 

amitom SeiZleba mxolod erTxel, erTad SevavsoT!}. 

TfojTwob; masw.-ma unda Tqvas marTebulad: `gverdebis an wveroebis 

raodenoba~. da ara ase: `gverdebis da wveroebis raodenoba~ 

(es iqneboda maTi jami, orive raodenoba erTad!). magram 

moswavleebs nu SevusworebT, am gansxvavebas mainc ver gaigeben. 

71/5/! `txsbgj! qbtvyfcj! `! uzvjmjb! Uv! nbsUbmj@~ masw.-i

- 184 - 

ambobs winadadebas, moswavlem ki swrafad unda upasuxos, 

tyuilia Tu marTali. amasTan, Tuki moswavle ityvis 

`marTalia~, amis dasabuTeba saWiro ar iqneba da moswavles es 

pasuxi an CaeTvleba (Tuki marTebulad daaskvna) an _ ara 

(Tuki SeeSala). magram Tuki moswavle ityvis `tyuilia~, man 

es pasuxi unda daasabuTos. winadadebebi yvela zogadi Sinaarsisaa. 

amitom dasabuTeba isaa, rom daasaxelos erTi magaliTi, 

romelic aCvenebs, rom zogadi winadadeba namdvilad mcdaria. 

winadadebaTa nimuSebi: 

1) pirveli Sesakrebi yovelTvis tolia meore Sesakrebisa. 

2) meore Sesakrebi zogjer tolia pirveli Sesakrebisa. 

3) pirveli Sesakrebi yovelTvis tolia jamisa. 

4) meore Sesakrebi zogjer tolia jamisa. 

5) pirveli Sesakrebi yovelTvis tolia nulisa. 

6) meore Sesakrebi zogjer tolia nulisa. 

7) pirveli Sesakrebi yovelTvis metia, vidre meore Sesakrebi. 

8) meore Sesakrebi yovelTvis naklebia, vidre pirveli Sesakrebi. 

9) pirveli Sesakrebi zogjer naklebia, vidre jami. 

10) jami zogjer naklebia, vidre pirveli Sesakrebi. 

11) jami zogjer metia, vidre meore Sesakrebi. 

12) jami arasdros araa naklebi, vidre pirveli Sesakrebi. 

13) pirveli Sesakrebi arasdros araa meti, vidre jami. 

14) jami ar SeiZleba iyos nulis toli. 

15) Sekrebisas SesakrebTa raodenoba yovelTvis metia, vidre erTi. 

16) Sekrebisas Sesakrebebi yovelTvis metia, vidre erTi. 

17) rac ufro didia kubi, miT ufro meti wvero aqvs. 

g 61. sjdywjtUwjt!UbwjtUbwjt!hbnplmfcb/!oblwUjt!tb{Swbsj/! 

72/1/ 60.9-is garCevisas misi magvari naxati unda iyos 

dafaze. Tuki klasSi araa feradi carcebi, maSin: _ dafis feri 

_ viTom wiTelia, carcis TeTri feri _ viTom mwvanea. rogor 

unda gagveferadebina? {mwvane da wiTeli; mwvane da mwvane; 

wiTeli da wiTeli; wiTeli da mwvane}.! 

72/2/!sjdywjtUwjt!UbwjtUbwjt!hbnplmfcb;!!9`9!!1-!!1`1!!1!!!eb! 

tywb/! am sagakveTilo monakveTs cota dro eyofa.

- 185 - 

jer ramdenime magaliTiT gavameorebinebT ricxvisTvis 

nulis mimatebas, nulisTvis ricxvis mimatebasa da ricxvisTvis 

nulis gamoklebas. 

dafazea rgolebis `dominoseburi~ cxraeuli. _ ramdenia? 

{cxra} gamovides aleko da gamoaklos cxra! (gamova da 

cxraves waSlis). ra dagvrCa? {nuli} risi toli iyo saklebi? 

{cxrisa} sxvaoba ra miviReT? {nuli}. maS, ricxvs rom 

TavisTavi gamovakloT, ras miviRebT? {nuls} diax, ricxvs 

gamoklebuli igive ricxvi tolia nulisa. aba ra, ricxvs rom 

TavisTavs gamovaklebT, xom aRaraferi ar dagvrCeba! 


wiTel CarCoSi (61.1) Caweril tolobebsa da utolobebs. 

bolos, yvela saklebs Semoxazaven mwvaned, risi tolicaa 

sxvaoba _ wiTlad. cxadia, sxvaoba mxolod tolobebSia, 

utolobebSi ki sxvaoba araa! 

72/3/! fsUjebo! frwtbnef! sjdywfcjt! Tftbcbnjt! epnjoptfcvs! 

xzpcbUb! hbombhfcb! zwfmb! Tftb[mfcmpcjt! bnpxvswjU-! spdb! 

vdopc!Tftb[mfcmpcbUb!sbpefopcb!bs!bSfnbufcb!5.t/! suraTi 

61.2. _ tomaraSi gviwyvia erTeulebi, oreulebi, sameulebi 

da oTxeulebi. mravlad, bevr-bevri gvaqvs. unda amoviRoT da 

davawyoT am kvadratebze, rogorc dominos qvebi. ginaxavT 

domino? ai amas hgavs domino. am suraTze ramdeni dominoa ukve 

Sevsebuli? {oTxi} ramden-ramdenia darCenili Sesavsebad? 

{eqvsi} diax, unda SevavsoT es eqvsi carieli dominos qva. 

oRond yvela domino sxvadasxva unda iyos! arcerTi dominos 

qva ar unda gameordes. ra aris pirvel dominoze? {erTi da 

erTi} meoreze? {erTi da ori} mesameze? {erTi da sami} 

meoTxeSi ra CavsvaT? {erTi da oTxi} CaxateT erTi da oTxi! 

mexuTe dominoze ra aris? {ori da ori} meeqvseSi ra CavsvaT? 

{ori da sami} diax, CaxateT ori da sami. Semdeg dominoze ra 

CavsvaT? {ori da oTxi} Semdeg dominoze ra CavsvaT? 

Tuki vinme ityvis `sami da ori~ an `sami da erTi~, masw.-i 

ikiTxavs: _ aba daakvirdiT, sami da ori ukve gvaqvs Tu ara? 

{gvaqvs ori da sami} es domino igivea, rac sami da ori!

- 186 - 

dominos qvas ginda ase daikaveb da ginda gadaabruneb! ese igi, 

sami da ori ar gvWirdeba, ukve gvaqvs! ra CavsvaT am dominoze? 

ra ar gvaqvs? {sami da sami} Semdeg dominoze? {sami da oTxi} 

bolo dominoze raRa CavsvaT? erTi gvaklia! {oTxi da oTxi}. 

-----------------------------------------------------------------------------------! 

JJJ.7/! nsbwbmlvUyfej/! nsbwbmlvUyfeUb! ebkhvgfcb;! tbnlvU. 

yfefcj-! pUylvUyfefcj! eb! btf! Tfnefh/! {phbebe! )bn!tjuzwjt! 

hbsfTf*! oblwUjt! tb{Swbsj/! lpncjobupsjlb! hfpnfusjbTj! `! 

sbnefoj! tbnlvUyfejt! )bo! tywb! oblwUjt*! ebobywbb! Tftb[mfcfmj! 

obyb{{f-! zwfmb! Tftb[mfcmpcjt! bnpxvswjU (kombinatoruli 

amocanebi yvela SesaZleblobis amowurviT da termini 

`SesaZlebloba~ klass ukve Sexvda meore xaziT). 

vaswavliT aTkuTxeds (varskvlaviseburs), misi daxatvas xeliT, 

mis gamorCevas sxvagvari nakvTebisagan. iseve rogorc sxva mravalkuTxedebi, 

aTkuTxedic Semogvaqvs zustad im gakveTilze, romelzec 

ariTmetikis xaziT pirvelad Semogvaqvs aTeuli da ricxvi aTi. 

Semogvaqvs zogadi termini `mravalkuTxedi~. zogadad mravalkuTxedis 

gverdebisa da wveroebis raodenobaTa toloba. mravalkuTxedis 

sazRvari Sedgeba gverdebisa da wveroebisagan. 

moswavles termini `nakvTi~ ukve mravaljer Sexvda saSinao Tu 

saklaso samuSaoebis Sesrulebisas. masw.-i mas pirdapir, TiTiT 

Cvenebis TanxlebiT ambobda. axlac asea. moswavleebi TandaTan 

SeeCvevian am termins. mis ganmartebas araviTari azri ara aqvs. misi 

zogadi mniSvnelobis gaazreba pirvelklasels arc moeTxoveba 

(gansxvavebiT, magaliTad, cnebisagan `mravalkuTxedi~ an `xazi~ 

SemdgomSi iqneba). 

am TavSi arsebiTad, axalia ori cneba: sazRvari da mravalkuTxedi. 

sxva yvelaferi naswavli sakiTxebis mxolod ganviTarebaa, 

arsebiTi siaxlis gareSe. mravalkuTxedi ganazogadebs anu gaaer- 

Tianebs ukve kargad cnobil samkuTxedebs, oTxkuTxedebs, xuTkuTxedebsa 

da ase Semdeg. mravalkuTxedi rTuli cnebaa imis gamo, 

rom Zalian zogadia. 

mere TvalsaCino CvenebiT varkvevT, wertili mravalkuTxedis 

SigniT mdebareobs Tu gareT. es sakmaod advilad irkveva. wvero?

- 187 - 

ai wvero ki arc SigniTaa da arc gareT. asevea sxva mravali 

wertilic. yvela isini mdebareobs mravalkuTxedis sazRvarze. 

mravalkuTxedisa da sazRvris Sesaxeb ix. zms $ 5. 

----------------------------------------------------------------------------------- 

72/4/!oblwUjt!tb{Swbsj/!njtj!Tfnpyb{wb!)hbnvrfcb*/!masw.-i dafaze 

sqematurad daxatavs or mezobel ezos, gamyofi RobiT. ezoebi 

Cans viTom zemodan, Robe mrude xazad Cans. _ ai viTom esaa ori 

ezo. es ezo petres ezoa, es ki _ vanosi. es ki viTom Robea 

ezoebs Soris (maCvenebeli joxis wvers gaayolebs `Robis~ 

xazs). es saxli visia? {vanosi} es xe vis ezoSi dgas? {vanosi} 

es xe? {petresi} es adgili visia? {vanosia} es? {petresia} es 

visia? vis ezoSia? (maCvenebel joxs daadebs zed Robis xazze) 

{arcerTisaa, arc vanosia da arc petresi} diax, es adgili 

ezoebs Soris sazRvarzea (am sityvis gamokveTiT). sazRvarze! 

Semdeg dafaze didad daixazeba raime uswormasworo nakvTi, 

romelsac aqvs nawilobriv mrude da nawilobriv texili 

sazRvari (aseTia, magaliTad, msr-Si 61.3-is wiTeli ukonturo 

nakvTi. masw.-ma sxva formisa unda daxazos). masw.-i maCvenebel 

joxs daadebs nakvTis romelime Siga wertils. _ es wertili 

am nakvTis Sigaa Tu gareT? {Sig} es? {esec Sigaa} es? 

{gareTaa} es? {Sigaa} es? {gareTaa} kargi, axla es wertili 

Sigaa Tu gareT? (miuTiTebs sazRvris romelime wertils). 

gareT aris? {ara} nakvTis SigniT aris? {ara} diax, arc Sigaa 

da arc gareT. sazRvarzea! 

_ axla daxedeT suraTs (61.3). jer daakvirdiT pirvel, 

vardisfer nakvTs. ra aris Savad gamuqebuli? {am nakvTis sazRvari} 

axla Tqven SeaferadeT lurjad wiTeli nakvTis sazRvari! 

(Semoavleben lurj xazs). aq kidev erTi nakvTis sazRvaria daxazuli. 

mxolod sazRvaria daxazuli, wyvetilad. Tqven mTlianad 

es nakvTi daxazeT da misi sazRvaric lurjad SeaferadeT! 

72/5/! ùzvjmjb! Uv! nbsUbmj@~! ` ix. 60.4. winadadebaTa 

nimuSebi: 1) kvadratis oTxive gverdi tolia. 

2) kvadratis xuTive gverdi tolia. 

3) eqvskuTxeds yovelTvis eqvsi gverdi aqvs.

- 188 - 

4) xuTkuTxeds yovelTvis xuTi wvero aqvs. 

5) oTxkuTxedis oTxive wvero wertilebia. 

6) samkuTxedis oTxive wvero wertilebia. 

7) marTkuTxedis yvela gverdi monakveTia. 

8) yvela kvadrati oTxkuTxedia. 

9) yvela marTkuTxedi oTxkuTxedia. 

10) yvela marTkuTxedi samkuTxedia. 

11) yvela marTkuTxedi rvakuTxedia. 

12) arcerTi marTkuTxedi araa xuTkuTxedi. 

13) arcerTi kvadrati araa eqvskuTxedi. 

14) yvela monakveTs ori bolo aqvs. 

15) monakveTis orive bolo wertilebia. 

16) monakveTis orive bolo wreebia. 

17) monakveTis samive bolo wertilebia. 

18) monakveTis oTxive bolo wertilebia. 

19) zog monakveTs mxolod erTi bolo aqvs. 

20) wris yvela wvero wertilebia. 

21) rgolis yvela wvero wertilebia. 

22) wris yvela gverdi monakveTebia. 

23) rgolis yvela gverdi monakveTebia. 

24) kvadratis oTxive gverdi wertilebia. 

25) SvidkuTxedis Svidive wvero monakveTebia. 

26) yvela oTxkuTxedis oTxive gverdi tolia. 

27) yvela rvakuTxedis rvave gverdi tolia. 

28) zogi samkuTxedis samive gverdi tolia. 

29) zogi cxrakuTxedis cxrave gverdi tolia. 

30) zogi cxrakuTxedis Svidi gverdi tolia. 

72/9/ ganvmartavT, rom dominos ujredebSi unda Caisvas an 

oreuli, an xuTeuli, an eqvseuli (tomridan).! 

g 62. bUfvmj/!sjdywj!bUj/!Tftb[mfcmpcfcj/!bUlvUyfej/! 

73/2/!bUfvmj/!sjdywj!bUj/!Tfebsfcb!obtxbwm!sjdywfcUbo/!yvelaferi 

keTdeba 12.1-is magvarad. erTaderTi, raki amgvari sakiTxebi 

ukve mravaljer damuSavda, sasurvelia tempis aCqareba.

- 189 - 

masw.-i klass dawyvilebiT erTmaneTs adarebinebs momzadebul 

grovebs (ix. saerTo miTiTebani I-2). dawyvilebiT aRmoCndeba, 

rom yvela aTeulSi nivTebis raodeonoba tolia. _ isrebisa da 

rkoebis raodenoba tolia. isrebi imdenivea, ramdenicaa rkoebi. 

xuTkuTxedebisa da wreebis raodenobac tolia. tolia, magram 

mainc ramdenia? {aTia} diax, esec aTeulia da esec. isrebis 

raodenobac aTia, rkoebisac. xuTkuTxedebi ramdenia? {aTi}. 


yvela aTeulSi aTi sagania. Semdeg masw.-i dafasTan gamoiyvans 


aTeulebi, TiTo moswavlem _ TiTo. Tanac, sxvadasxvanairi 

aTeulebi, sxvisi daxatuli ar gaimeoron. mosalodnelia, rom 

moswavleebi gaimeoreben im sxvadasxva ganlagebas, romlebic 

masw.-is mier iyo momzadebuli. masw.-i miuTiTebs moswavleTa 

mier daxatul aTeulebze: _ aqac aTi rgolia, aqac, aqac, aqac. 

romelia yvelaze lamazi? (gamoiTqmeba Sexedulebani) me mgoni, 

es jobia yvelas (da Semoxazavs `dominosebr~ aTeuls). 

Semdeg aTs Sedardeba ukve naswavli ricxvebi: erTi, ori, sami, 

... cxra. amisaTvis dafaze `dominoseburad~ daewyoba ramdenime 

saTanado grova (aTeuli da eqvseuli, aTeuli da cxraeuli, 

aTeuli da rvaeuli) da dawyvilebiT Sedardeba erTmaneTs. 

_ romeli raodenobaa meti _ aTi Tu eqvsi? {aTi} ratom? 

{mogvrCa...} romelia naklebi _ aTi Tu cxra? {cxra} ratom? ... 

suraTi 62.1. _ ra aris aq varskvlavSi? {aTi patara Savi 

wre} risi aTeulia aq? {wreebis aTeuli} axla Tqven mis 

qvemoT daxazeT lurji wreebis aseTive aTeuli! aq kidev xatia 

sxvadasxva grovebi. jer burTebis grovaa. igi SeaerTeT mis 

Sesabamis raodenobasTan, qvemoT! _ da ase gameordeba 

fanqrebis, foTlebis, fantelebisa da sxivebis grovebzec. 

_ erT xelze ramdeni TiTi gvaqvs? meore xelze? ramdeni 

TiTi gvaqvs orive xelze? miCvene aTi TiTi! daakakune aTjer! 

daukari taSi aTjer! vin mixvdeba, ramdeni TiTi gvaqvs orive 

fexze erTad? {aTi} saidan ici? {erT fexze gvaqvs xuTi, 

meorezec xuTi, sul aTia}. diax, xuTi da xuTi _ aTia. aba

- 190 - 

daakvirdiT am aTeuls (miuTiTebs dafaze daxatul `dominosebr~ 

aTeulze). rogoraa aq dawyobili aTeuli? {zemoT xuTi 

da qvemoTac xuTi, xuTi da xuTi}. 

73/3/!obybufcjt!hbombhfcb!bo!obyb{jt!hbgfsbefcb!zwfmb!Tftb. 

[mfcmpcjt! bnpxvswjU-! spdb! vdopc! Tftb[mfcmpcbUb! sbpef. 

opcb! bs! bSfnbufcb! 7.t/! suraTi 62.2-is erTi wre dafazec 

daixazeba. _ ramden nawiladaa dayofili wreebi? {sam nawilad} 

ramdeni wrea gaferadebuli? {sami} ramdenia gasaferadebeli? 

{xuTi} diax, TviTeuli nawili unda iyos an wiTeli an ruxi. 

mxolod (am sityvis gamokveTiT) es ori feri unda gamoviyenoT! 

pirveli wre rogoraa gaferadebuli? {samive wiTelia} 

meore? {ori nawili wiTelia da erTi _ ruxi} mesame? {isic 

isea, magram sxva nawilia ruxi} axla ganvagrZoT gaferadeba. 

rogor gavaferadoT pirveli gauferadebeli wre? ... 

danarCeni yvelaferi iseve, rogorc 61.2-Si.! 

73/4/! bUlvUyfej! )wbstlwmbwjtfcvsj*/! njtj! ebybuwb! yfmjU/! 

njtj!tb{Swbsj/!!_ daaTvaliereT suraTi (62.3). ras hgavs wiTeli 

nakvTi? {varskvlavs} ramdenkuTxedia igi? {aTkuTxedia} ramdeni 

gverdi aqvs? daTvaleT! {aTi} ramdeni wvero aqvs? {aTi}. 

Semdeg daxazaven da lurjad an mwvaned gaaferadeben 

danarCen aTkuTxedebs (romlebic mxolod konturuladaa moxazuli). 

yvelas sazRvars wiTlad Seaferadeben. 

bolos, masw.-i klass aswavlis varskvlavis xeliT daxatvas 

_ xelis uwyveti moZraobiT: erTi wverodan rom iwyeba da 

bolos imave wveroSi rom mTavrdeba. masw.-i dafaze Zalian 

didad akeTebs am moZraobas. moswavleebi 62.3-is cariel 

adgilas imeoreben. 

73/5/!Ùwbmjt!ebybnybnfcb~/ ix. 15.4. 

g 63. `nblmfcj~/!Tftb[mfcmpcboj/!wbstlwmbwjtfc/!bUlvUyfefcj/ 

74/2/!ufsnjoj!`nblmfcj~/!dafaze erT rigSi daxatuli `dominoseburi~ 

xuTeuli, cxraeuli, Svideuli, aTeuli, oTxeuli 

(rgolebisa). _ xuTs gamovakloT ori. cxras gamovakloT ori. 

Svids gamovakloT ori. aTs gamovakloT ori. oTxs gamovak-

- 191 - 

loT ori. (gamodis moswavle da waSlis saTanado Sesabamisi 

raodenobis rgolebs). 

_ romeli ricxvebi iyo saklebebi? {xuTi, cxra, Svidi, aTi, 

oTxi} romel ricxvs vaklebdiT am saklebebidan? {ors} 

ricxvs, romelic unda gamovakloT, maklebi hqvia. maklebi. maS, 

saklebidan vaklebT maklebs. 


saklebi? {cxra} diax, romelsac vaklebT (TiTs daadebs 

cxrians), isaa saklebi. romelia aq maklebi? {xuTi} Sesakrebebi 

rogor gvqonda? pirveli Sesakrebi da meore Sesakrebi. saklebi 

ki ase ar SeiZleba. pirveli saklebi da meore saklebi _ 

uazrobaa, ar SeiZleba! saklebi mxolod erTia! maklebi ki 

ramdenia? {erTi} diax, saklebic erTia da maklebic erTi! am 

tolobaSi es oTxi ra aris? {sxvaoba} es xuTi? {maklebi} es 

cxra? {saklebi}. 


cxrilSi (63.1) Caweril tolobebs. maklebebs wiTlad 

Semoxazaven. 

_ axla daukvirdiT: romelia xolme meti _ saklebi Tu 

maklebi? {saklebi} SeiZleba Tu ara, rom saklebi maklebis 

toli iyos? {SeiZleba, magaliTad: 5–5=0 _ es toloba dafaze 

daiwereba} SeiZleba Tu ara, rom maklebi meti iyos, vidre saklebi? 

{ar SeiZleba} diax, maklebi meti ar SeiZleba iyos! 

74/3/!fsUjebo!frwtbnef!sjdywfcjt!Tftbcbnjt!epnjoptfcvs!xzpcb. 

Ub! hbombhfcb! zwfmb! Tftb[mfcmpcjt! bnpxvswjU-! spdb! vdopc! 

Tftb[mfcmpcbUb! sbpefopcb! bs! bSfnbufcb! 7.t/! yvelaferi iseve, 

rogorc 61.2-Si. mxolod SesaZleblobaTa raodenobaa meti. 

74/4/! tywbebtywbhwbsj! wbstlwmbwjtfcvsj! bUlvUyfefcj/! nbUj! 

tb{Swbsj/!_ suraTze (63.3) daxazulia sxvadasxvagvari nakvTebis 

sazRvari. mxolod sazRvaria daxazuli, Tqven ki 

gaaferadeT! 

saTiTaod daTvlian wveroebs da darwmundebian, rom yvela 

nakvTi _ aTkuTxedia. bolos wiTlad Seaferadeben sazRvrebs. 

74/5/!èbnbmvmj!djgsjt![fcob~/ ix. 44.4.

- 192 - 

g 64. {fqjsj!Tflsfcb/!nfbUf/!nsbwbmlvUyfej!`!ufsnjoj/ 


75/2/!{fqjsj!Tflsfcb!bUjt!gbshmfcTj/!masw.-is daxmarebiT bav- 

Svebi Slian mTw-s `10~-ze (Canaweri 10 jer ar Semogvaqvs _ 

yvelaferi mxolod sityvierad warmoiTqmis!). _ aq ramdeni 

wrea? {aTi} aq ki ramdeni wrea? {aqac aTi} aTi wre rogor 

aris aq dawyobili? {xuTi da xuTi}. axla davTvaloT wreebis 

raodenoba aq! ramdeni wrea? {aTi} aq rogor aris dawyobili 

aTi wre? {rva da ori} aq ramdeni wrea? {aqac aTi} aq rogor- 

Ra aris dawyobili aTi wre? {eqvsi da oTxi} (da ase Semdeg). 


vapasuxebinebT: _ rogor iyo dalagebuli aTi wre? 

{eqvsi da oTxi} kidev rogor iyo? {xuTi da xuTi} kidev?.. 


Semdeg daiTvleba, rogoraa dawyobili aTi TviTeul 

`jamSi~ (elifsSi). am TvalsaCinoebis daxmarebiT gamoTvlian 

yvela Sesabamis jams: 9+1, 1+9, 8+2, 2+8 da ase Semdeg. 

garda amisa, gzadagza daismis amdagvari SekiTxvebi: _ cxras 

ramdeni unda mivumatoT, rom aTi gaxdes? erTs ramdeni unda 

mivumatoT, rom aTi gaxdes? rvas ramdeni unda mivumatoT, rom 

aTi gaxdes? ors ramdeni unda mivumatoT, rom aTi gaxdes? 

Svids ramdeni unda mivumatoT, rom aTi gaxdes? sams ramdeni 

unda mivumatoT, aTi rom gaxdes? eqvss ramdeni unda 

mivumatoT, rom aTi gaxdes? oTxs ramdeni unda mivumatoT, aTi 

rom gaxdes? xuTs ramdeni unda mivumatoT, aTi rom gaxdes? 

75/3/!sjhpcjUj!sjdywj!nfbUf/!qjswfmjebo!nfbUfnef!sjhpcjUj! 

sjdywfcjt! eblbwTjsfcb! obtxbwm! tjwsdjU! njnbsUfcfcUbo/!! 

ganvixilavT nakvTebis rigs (suraTi 64.2) da davsvamT amgvar 

SekiTxvebs: _ romeli nakvTia marcxnidan (an: marjvnidan) 

meSvide? mecxre? meaTe? kvadrati marcxnidan meramdenea? 

marjvnidan? (da ase Semdeg). 

Semdeg ganvixilavT mTw-is ydis suraTs da movZebniT, vin 

aris meaTe. Tan gavimeorebT, vin aris meore, mexuTe ... ... 

75/4/!nsbwbmlvUyfej!`!{phbej!ufsnjojt!Tfnpubob/!

- 193 - 

dafaze didad daixazeba raime uswormasworo, Cazneqili 

aTkuTxedi da mis gverdiT TiTqmis aseTive nakvTi, oRond 

romelsac erTi kuTxe momrgvalebuli aqvs. 

_ daTvaleT, ramdeni wvero aqvs marcxena nakvTs {aTi} ra 

nakvTia igi? {aTkuTxedi} dauTvlelad, swrafad mipasuxeT: 

ramdeni gverdi eqneba mas? {aTi} saidan iciT? {gverdebisa da 

wveroebis raodenoba tolia} diax, esaa aTkuTxedi. daakvirdiT 

axla marjvena nakvTs. riTi gansxvavdeba igi am aTkuTxedisagan? 

{erTi kuTxe aqvs momrgvalebuli} aris Tu ara es nakvTi 

aTkuTxedi? {ara} ratom? {aTkuTxeds arsad momrgvalebuli ar 

unda hqondes} aris Tu ara es nakvTi cxrakuTxedi? {ara, arc 

cxrakuTxedia!} SeadareT erTmaneTs am ori nakvTis sazRvari. 

aTkuTxedis sazRvari Sedgeba mxolod gverdebisagan. gverdi ra 

nakvTia? {monakveTia} diax, aTkuTxedis sazRvari mxolod 

monakveTebisagan Sedgeba. ramdeni monakveTisagan? {aTi 

monakveTisagan} am marjvena nakvTis sazRvari ki rogoria? {ar 

Sedgeba mxolod monakveTebisagan, erT adgilas mrudea!} diax, 

ai aq araa monakveTi! swored amitom araa es nakvTi arc aTkuTxedi, 

arc cxrakuTxedi da arc rvakuTxedi! 

_ Tu mixvdebiT, rogor nakvTs hqvia mravalkuTxedi? 

{romelsac iseTi sazRvari aqvs, romelic mxolod 

gverdebisagan, mxolod monakveTebisagan Sedgeba} samkuTxedebi, 

kvadratebi, marTkuTxedebi, aTkuTxedebi, cxrakuTxedebi, 

rvakuTxedebi _ yvela eseni mravalkuTxedebia! es marjvena 

nakvTi ki ar aris mravalkuTxedi! axla Tqven miTxariT: aris 

Tu ara mravalkuTxedi wre? {ara} rgoli? {ara} wertili? 

{ara} monakveTi? {ara} oTxkuTxedi? {ki}. 

Semdeg daiwyeben 64.3-is Sesrulebas. gakveTilze cota 

gaaferadon, danarCeni _ saSinao davalebad. 

75/5/! `lsjbybt! xjxjmfcj~ _ axla me mogiyvebiT zRapars. 

iyo da ara iyo ra, erT ezoSi lamazi Wreli dedali cxovrobda, 

saxelad kriaxa. erTxel mokruxda da sabudarSi kvercxebi 

dado. aba, erTad daviTvaloT, ramdeni kvercxi dado: er- 

Ti, ori, sami... (erTad iTvlian)... aTi! kriaxam aTi kvercxi, da-

- 194 - 

do! daajda kvarcxebs da aTbobs, aTbobs. gavida xani. uceb kriaxas 

erTi kvercxidan fxakuni moisma, Cik-Cik, gatexa Signidan 

wiwilam kvercxi da gamoxta, gamoiCeka! wiwila saxelad wiaqa! 

kvercxi ramdeni darCa? {cxra} wiaqas jer aqeT-iqiT mixedvac 

ki ver moeswro, rom meore kvercxidanac Cik-Cik da wiwila gamoiCeka! 

erTi da erTi _ ori wiwila. kvercxi ramdeni darCa? 

{rva} wiwilebi ki dawivwiveben da sxvebs ucdian. Cik-Cik! meramdene 

wiwila gamoiCeka? {mesame!} ori da erTi _ sami wiwila. 

yvelam erTad gaimeoreT! {ori da erTi _ sami wiwila} 

kvercxi ramdeni darCa? {Svidi} Cik-Cik! meramdene wiwila gamoiCeka? 

{meoTxe!} sami da erTi _ oTxi wiwila. kvercxi ramdeni 

darCa? {eqvsi} gaimeoreT! {sami da erTi _ oTxi wiwila, kvercxi 

ki eqvsi darCa} Cik-Cik! axla Tqven TviTon TqviT erTad! 

{mexuTe wiwila gamoiCeka! oTxi da erTi _ xuTi wiwila, kvercxi 

ki xuTi darCa} dawivwiveben sabudarSi yviTeli fumfula 

wiwilebi. uceb Cik-Cik! ra unda TqvaT? {meeqvse wiwila gamoi- 

Ceka! xuTi da erTi _ eqvsi wiwila, kvercxi ki oTxi darCa} 

Seusxdnen kriaxas frTebis qveS. Cik-Cik! {meSvide wiwila gamoiCeka! 

eqvsi da erTi _ Svidi wiwila, kvercxi ki sami darCa} 

meSvidem martom daiwyo aqeT-iqiT kunkuli. Cik-Cik! {merve wiwila 

gamoiCeka! Svidi da erTi _ rva wiwila, kvercxi ki ori 

darCa} gareT ori wiwilaa, dawivwiveben. frTebis qveS ki ramdeni 

wiwilaa? {eqvsi} or axalsac mounda kriaxas frTebis 

qveS, SeZvrnen. ramdeni wiwilaa frTebis qveS? {rva} magram iq 

ver moisvenes. sami wiwila gareT gamovarda. ramdeni wiwila 

darCa frTebis qveS? {xuTi} sami ki gareT dawivwivebs. uceb 

Cik-Cik! {mecxre wiwila gamoiCeka! rva da erTi _ cxra wiwila, 

kvercxi ki erTi darCa} mecxre maSinve kriaxas eca da frTis 

qveS SeZvra. ramdeni wiwilaa frTebis qveS? {eqvsi} gareT ramdenia? 

{sami} am samidan yvelaze mousvenar wiaqa iyo. eca wiaqa 

erT darCenil kvecxs da pawia niskartiT Catexa. magram vai, 

rogor SeeSinda! Cik-Cik! amoxta wiwila! {meaTe wiwila gamoi- 

Ceka! cxra da erTi _ aTi wiwila, kvercxi ki aRar darCa} 

ramdeni wiwilaa frTebis qveS? {eqvsi} gareT ramdenia? {oTxi}

- 195 - 

yvelaze mousvenari romelia? {wiaqa} wiaqa romeli wiwilaa? 

{yvelaze pirvelad rom gamoiCeka, yvelaze ufrosi!} isev frTebis 

qveS SeZvra wiaqa, daiwyo iq xvancali da ori wiwila gareT 

gamoagdo. ramdeni wiwilaa gareT? {xuTi} frTebis qveS? 

{xuTi} xuTi da xuTi _ aTia. erTi frTis qveS sami wiwilaa. 

ramdeni wiwilaa meore frTis qveS? {ori} gareT ramdenia? 

{xuTi} SescivdaT im xuT wiwilas da isinic kriaxas frTebis 

qveS SeZvrnen, magram iqidan wiaqa gamovarda! ramdeni wiwilaa 

frTebis qveS? {cxra} gareT? {erTi} cxra da erTi _ aTia. 

erTi frTis qveS oTxi wiwilaa. ramdeni wiwilaa meore frTis 

qveS? {xuTi} gareT ramdenia? {erTi}. 

da ase Semdeg. masw.-s SeuZlia maSin daamTavros, roca 

saWirod miiCnevs. 

g 65. {fqjsj!hbnplmfcb/!hbnfpsfcb/!xwfspfcj-!hwfsefcj/ 

76/1/ 64.8 saintereso amocanaa. misi amoxsna dafazec unda 

daixatos lamazad. 

76/2/! {fqjsj! hbnplmfcb! bUjt! gbshmfcTj/! jer gameoreba: 

yvelaferi gakeTdeba igive, rac 64.1-Si, oRond swrafad. Semdeg 

mTw-s daxmarebiT moswavleebi iangariSeben yvela sxvaobasac: 

10 _ 1, 10 _ 2 da ase Semdeg. 

_ ramdenia aTs gamovakloT erTi? aTs gamovakloT cxra? 

aTs gamovakloT ori? aTs gamovakloT rva? aTs gamovakloT 

sami? aTs gamovakloT Svidi? aTs gamovakloT oTxi? aTs 

gamovakloT eqvsi? aTs gamovakloT xuTi? 

_ amixseni, rogor mixvdi, ramdenia aTs gamovakloT Svidi 

{aTi wre dawyobilia Svidad da samad; aTs rom gamovaklebT 

Svids, darCeba sami}. 

_ aTs ramdeni unda gamovakloT, rom darCes eqvsi? aTs 

ramdeni unda gamovakloT, rom darCes oTxi? aTs ramdeni unda 

gamovakloT, rom darCes Svidi? aTs ramdeni unda gamovakloT, 

rom darCes sami? 

76/3/ Ubwj!8/3.jt!hbnfpsfcb!eb!Tfkbnfcb/! 

yvelaferi iseve, rogorc 61.2-Si. erTi gansxvavebaa: axla

- 196 - 

unda SevavsoT ara `dominoebi~, aramed uZravi oTxkuTxedis 

ujredebi. amitom ganlageba èrTi, ori sami~ da èrTi, sami, 

ori~ _ sxvadasxvaa. Tanac, erT oTxkuTxedSi raodenobaTa gameoreba 

ar SeiZleba. magaliTad, ar SeiZleba èrTi, erTi, ori~. 

76/4/! {phbebe! nsbwbmlvUyfejt! tb{Swbsj! `! xwfspfcj-! hwfs. 

efcj/! _ risgan Sedgeba mravalkuTxedis sazRvari? {gverdebisagan} 

gverdi ra nakvTia? {monakveTi} ese igi, risgan Sedgeba 

mravalkuTxedis sazRvari? {monakveTebisagan} mxolod monakve- 

Tebisagan, mravalkuTxeds meti araferi ar SeiZleba hqondes! 

ramdeni monakveTisagan Sedgeba mravalkuTxedis sazRvari? {ramdenkuTxedicaa, 

imdeni monakveTisagan} magaliTad, aTkuTxedi? 

{aTi monakveTisagan} yvelanairi aTkuTxedi? sulerTia, rogori? 

{ki, sulerTia, aTkuTxedi rogoria, misi sazRvari _ mainc 

aTi gverdisgan Sedgeba}. wvero ramdeni aqvs mravalkuTxeds? 

{ramdenkuTxedicaa, imdeni}. sad mdebareobs mravalkuTxedis 

wvero? (Tuki ver upasuxeben, daemateba SekiTxva: sad mdebareobs 

_ mravalkuTxedis Sig Tu gareT?) {mis sazRvarze} yvela 

wvero sazRvarzea Tu zogi? {yvela sazRvarze mdebareobas}. 

_ aris Tu ara mravalkuTxedi pirveli nakvTi naxazze 

(65.3)? {aris} ramdenkuTxedia? {SvidkuTxedi} SeaferadeT 

mwvaned misi sazRvari, xolo wveroebi lurjad moniSneT! 

Semdeg Seasruleben iseTsave davalebas, rac iyo 64.3-is 

bolos. 

76/5/!`txsbgj!qbtvyfcj~!`!bUjt!gbshmfcTj!{fqjsj!Tflsfcb. 

hbnplmfcb/! 

76/8/!yuradReba unda mieqces imas, rom erTi zoli yvelgan 

meores `zemodanaa~ da zolebi ise unda gaferaddes, rom es ar 

dairRves! Semowmebisasac es unda Semowmdes. 

g 66. hbnfpsfcb/!hbnplmfcjt!dysjmj/!nsbwbmlvUy/!ebkhvgfcb/! 

77/1/! 65.8-Si SesaZleblobebia: wiTeli, wiTeli; wiTeli, 

mwvane; lurji, wiTeli; lurji, mwvane; Savi, wiTeli; Savi, 

mwvane. bolos vikiTxoT: _ maS ramdeni sxvadasxva SesaZlebloba 

yofila? {eqvsi}.

- 197 - 

77/2/!Ubwj!9/2.jt!Tfkbnfcb!eb!hbnfpsfcb/!hbnplmfcjt!dysjmjt! 

eblbwTjsfcb! ufsnjofcUbo! `tblmfcj-! nblmfcj-! tywbpcb~/! 

gamoklebis cxrili 65.7. 66.1 ukve Sevsebulia saSinao 

davalebaSi. Semowmdeba da Secdomebi kargad gairkveva da 

gaswordeba. Semdeg gairkveva, gamoklebis cxrilSi Caweril 

ricxvTagan romelia saklebi, romelia maklebi da romelia 

sxvaoba. amisaTvis masw.-i dafaze dawers gamoklebis erT 

magaliTs cxrilidan, magaliTad, 9 _ 3 = 6. 

_ romelia aq saklebi? gamoklebis cxrilSi sad unda CavweroT 

igi? romelia aq maklebi? cxrilSi sad unda CavweroT 

igi? risi tolia aq sxvaoba? cxrilSi sad unda CavweroT igi? 

moZebneT cxrilis mesame sveti, am svetis mesame ujredi, mesame 

striqonis gadakveTis ujredi. am ujredSi ra weria? ratom? 

romelia aq saklebi? maklebi? sxvaoba? (da kidev ori ujredi 

aseve ganixileba). 

Semdeg mTw-s daxmarebiT Sesruldeba aTis farglebSi zepiri 

Sekreba-gamokleba. 

------------------------------------------------------------------------------------ 

JJ.9/!tbnjebo!bUbnef!sjdywfcjt!ebTmb!TftblsfcUb!kbnfcbe!`! 

zwfmb! Tftb[mfcmpcjt! bnpxvswjU/ esaa kombinatorikisa da 

ariTmetikis saerTo sakiTxi, Zalian mniSvnelovani. saerTo Ziebis 

Semdeg dafaze Camoiwereba yvela SesaZlebloba tolobaTa saxiT. 

klasi Tan kiTxulobs. 

psnprnfefcjboj! Tflsfcb.hbnplmfcb! sjdywUb! ljcf{f/ 

savarjiSoebi mxolod cxris farglebidan unda SeirCes, radganac 

aTis cifruli Canaweri jer ar gviswavlebia da ricxvTa kibeze mas 

jer ar wavakiTxebT. 

maSasadame, viwyebT ormoqmedebian gamoTvlebs. swored mravalmoqmedebian 

gamoTvlebs aadvilebs ricxvTa kibe, Torem 

erTmoqmede-bianebis masze Sesrulebas didi azri arca aqvs. klass 

TandaTan zepiradac vasrulebinebT advil ormoqmedebian 

gamoTvlebs. Secdomisa an gaWirvebis SemTxvevaSi kvlav ricxvTa 

kibes viSveliebT. 

garda amisa, grZeldeba `-iT metoba~ da `-iT nakleboba~

- 198 - 

umartives sayofacxovrebo amocanebSi. 

----------------------------------------------------------------------------------- 

77/3/!pUyjt-!yvUjt-!tbnjtb!eb!psjt!ebTmb!zpwfmhwbs!Tftbl. 

sfcUb! kbnfcbe/! mTw gadaiSleba `4~-ze. _ radara ricxvebis 

jamis tolia oTxi? {samisa da erTis} kidev? {orisa da oris} 

kidev Tu aris sxva SesaZlebloba? {erTi da sami} ara, es ar 

gvinda, erTi da sami xom igivea, rac sami da erTi! sxva 

SesaZlebloba Tu aris? {oTxi da nuli} kidev Tu aris sxva 

SesaZlebloba? {ara} ese igi, rogor da rogor SegviZlia 

davSaloT oTxi SesakrebTa jamad? {sami da erTi; ori da ori; 

oTxi da nuli}. axla daxedeT suraTs (66.2). jer sul 

marcxniv. xesaviT, fiWviviT raRacaa daxazuli. zemoT, kenwerosTan 

ra aweria? {oTxiani} diax, es oTxia jami, aqeT-iqiT 

totebze ki Sesakrebebia! waikiTxeT, radara Sesakrebebia! {ori 

da ori; sami da erTi; oTxi da nuli}. esaa oTxis oris daSla 

yovelgvar SesakrebTa jamebad. sxva SesaZlebloba Tu aris 

daSlisa? {ara}. 

mTw gadaiSleba `5~-ze. yvelaferi gakeTdeba iseve, rogorc 

zemore (oTxisaTvis), oRond oTxis nacvlad daiSleba xuTi. 

erTi gansxvavebiT _ msr-s 66.2-Si meore, xuTianis `fiWvi~ 

TviTon moswavleebma unda Seavson. 

Semdeg mTw gadaiSleba `1-2-3~-ze. yovelgvar SesaZlo 

SesakrebTa jamebad daiSleba sami da ori. msr-s 66.2-Si samianisa 

da orianis `fiWvebsac~ TviTon moswavleebi Seavseben. 

77/4/! zpwfmhwbsj! nsbwbmlvUyfejt! hwfsefcjtb! eb! lvUyffcjt! 

sbpefopcbUb! upmpcb! `! ebxzwjmfcjU/! nsbwbmlvUyfeUb! 

ebkhvgfcb;!tbnlvUyfefcj-!pUylvUyfefcj!eb!btf!Tfnefh/! 

dafaze daxazulia didi uswormasworo Cazneqili aTkuTxedi 

(rogorc yovelTvis, mTlianad gaferadebulia, SeiZleba, 

TeTradve). gairkveva, rom es _ aTkuTxedia. _ amis gverdebis 

raodenobaa meti Tu wveroebisa? {arcerTia meti, es 

raodenobani tolia, gverdebi imdenivea, ramdenicaa wvero} 

rogor SegviZlia gavigoT, raodenobani tolia Tu ara? ai sul 

Tavidan rom viswavleT grovebis Sedareba! {dawyvilebiT} diax,

- 199 - 

gamovides tarieli da daawyvilos amis gverdebi da wveroebi. 

erTi gverdi _ erTi wvero... naxatze rogor vawyvilebT 

xolme? {xazebiT vaerTebT} (moswavle gamova da daawyvilebs 

xazebiT SeerTebiT) Tu mogvrCa romelime? {ara, raodenobani 

tolia}. axla daxedeT suraTs (66.3), Tqven rom gaaferadeT. 

ramdeni nakvTia aq daxazuli? {rva} daakvirdiT _ rvave 

mravalkuTxedia Tu ara? {ki} yvelam daawyvilos maTi wveroebi 

maT gverdebTan, rogorc tarielma daawyvila dafaze! 

Tuki romelime gauWirdebaT, dafaze daixazeba da 

dawyvildeba. 

_ radara nakvTebad iyofa mravalkuTxedebi? radara nakv- 

Tebad SeiZleba davajgufoT mravalkuTxedebi? {samkuTxedebi, 

oTxkuTxedebi, xuTkuTxedebi, eqvskuTxedebi, SvidkuTxedebi...} 

gairkveva, suraTze ramdenia TviTeuli am jgufis 

mravalkuTxedi. 

77/5/!`txsbgj!qbtvyfcj~!`!bUjt!gbshmfcTj!{fqjsj!Tflsfcb. 

hbnplmfcb/! 

77/7/!SeiZleba gakeTdes dawyvilebiT an wreebis gadaTvliT. 

----------------------------------------------------------------------------------- 

J.:/! viwyebT sul axal Temas:! UwbmtbDjope! npdfnvm! sbpef. 

opcbUb!ebUwmb!kvgUfcjU/!!jer yvelaze advili SemTxvevebia 

unaSTo. yvelaferi mxolod zepirad keTdeba, saTanado CanawerTa 

gareSe. kidev erTxel damuSavdeba terminebi `yoveli~, `TviTeuli~. 

jufTis Sesaxeb ix. zms $ 3. Tumca, zogad termins `jufTi~ 

klasSi arcerTxel ar vaxsenebT. igi mxolod masw.-isTvisaa. klasSi 

vityviT mxolod konkretulad: oTxeuli, aTeuli, cxraeuli, 

wyvili da sxva romelic dagvWirdeba. 

jer-jerobiT (Tav 12-mde), arsad ar daismis sakiTxic ki imis 

Sesaxeb, Tu sul ramdeni sagania grovaSi! jer mTavaria is, rom 

moswavlem iswavlos TviT jufTebis daTvla, ese igi, man mTliani 

grova-jufTi erTian sagnad unda CaTvalos. amas sakmao muSaoba 

sWirdeba. 

karga xnis ganmavlobaSi grovebi jufTebad ewyoba ise, rom 

naSTi ar morCes. amisaTvis saWiroa, rom masw.-i gamudmebiT aqcevdes

- 200 - 

yuradRebas: mis mier momzadebul grovaSi unda iyos imdeni nivTi, 

rom unaSTod daewyos. magaliTad, roca moswavles avalebs 

oTxeulebad dawyobas, grovaSi SeiZleba iyos 8, 12, 16, 20, 24, 28, 

32 nivTi oTxis jeradi raodenobisa, Tan ise, rom miviRoT ori 

an meti grova! roca moswavles avalebs Svideulebad dawyobas, 

grovaSi SeiZleba iyos 14, 21, 28, 35 nivTi Svidis jeradi 

raodenobisa (saerTo raodenoba ormocze meti ar iyos). 

Tavidan mocemuli grova areulia, an grova sul araa mocemuli 

(magaliTad, roca unda daixatos). magram sasurvelia, rom 

jufTebi (sameulebi, Svideulebi da ase Semdeg) `dominoseburad~ 

daewyos. ase ufro advili aRsaqmeli iqneba saboloo suraTi. 

---------------------------------------------------------------------------------- 

g 67. hspwjt!ebxzpcb!kvgUfcbe/!Tflsfcb!sjdywUb!ljcf{f/! 

78/2/! ojwUUb! hspwjt! ebxzpcb! kvgUfcbe;! pUyfvmfcbe-! 

tbnfvmfcbe-! yvUfvmfcbe! eb! tywb! )UwjU! bn! kvgUfcjt! ebUwmjt! 

hbsfTf*/!keTdeba Zalian advili savarjiSoebi. sxvadasxva 

moswavleni rig-rigobiT awyoben nivTTa grovebs jufTebad. 

magaliTad: _ gamovides gurami da es kvadratebis grova 

daawyos ise, rom sul oTxeulebi iyos! 

gurami gamova da grovidan arCevs oTxeulebs: daawyobs erT 

oTxeuls, mere meore oTxeuls da ase Semdeg, ramdenic gamova. 

jer-jerobiT ar vaTvlevinebT, Tu sul ramdeni oTxeulia. _ 

Tornike, ramden-ramdenad daawyo guramma kvadratebis grova? 

{oTxeulebad} oTxeulebad daawyo (gamokveTiT). 

Semdeg iaTamze daawyobs Cxirebis grovas ise, rom sul 

Svide-ulebi iyos. _ vaxo, ramden-ramdenad daawyo iaTamzem 

Cxirebis grova? {Svideulebad} Svideulebad daawyo 

(gamokveTiT). da ase Semdeg. 

TviTeuli dawyobis Semdeg masw.-i dasvams SekiTxvas: 

_ es kubebi axla ufro metia Tu Tavidan iyo ufro meti? 

{axla imdenivea, ramdenic Tavidan iyo} ratom? {grova igivea, 

marto davawyveT sxvanairad!}. 

78/3/!psnprnfefcjboj!Tflsfcb!sjdywUb!ljcf{f/! 

jer swrafad gavameorebinebT Sekreba-gamoklebas ricxvTa

- 201 - 

kibeze. Semdeg daviwyebT ormoqmedebian Sekrebas ricxvTa kibeze. 

rogorc iTqva, savarjiSoebi mxolod cxris farglebidan 

unda SeirCes, radganac aTis cifruli Canaweri jer ar 

gviswavlebia da ricxvTa kibeze mas jer ar wavakiTxebT. 

savarjiSoTa nimuSebi: _ iangariSe ricxvTa kibeze 2+3+2. 

joxis wveri ar moacilo kibes, sanam pasuxamde ar mixval! 

axla besom (Zlierma moswavlem) adgilidan iangariSos 5+1+2. 

merabma ki ricxvTa kibeze Seamowmos, besos xom ar SeSlia. 

erTi moswavle moqmedebas did saklaso ricxviT kibeze 

asrulebs, sxvebi _ Tav-TavianT mTw-Si. 

78/4/! zwfmb! Tftb[mfcmpcjt! bnpxvswjU;! sbnefoj! tbnlvUyfej! 

)lwbesbuj!bo!nbsUlvUyfej*!Tfj[mfcb!ebwjobypU!obyb{{f!)bsbv. 

nfuft!frwtjtb*/!nbUj!dbm.dbmlf!bnpybuwb/!ganvixilavT naxazebs 

67.3. viwyebT marcxnidan, pirvelia naZviviT nakvTi. _ ra nakv- 

Tebisagan Sedgeba igi? {samkuTxedebisagan} ramdeni samkuTxedisagan? 

{samisagan} amoxateT yvela, oRond cal-calke! 

masw.-i am dros, Cveulebriv, merxebs Soris Camoivlis da 

amowmebs, rom moswavleTa mier daxatuli samkuTxedebi 

daaxloebiT imxelebi iyos, ramxelebicaa naZvisebr nakvTze. 

Semdeg gadavdivarT meore Sedgenil nakvTze. _ ramdeni 

samkuTxedis danaxva SegiZliaT aq? ar gamogrCeT Camalulebi! 

{samisa} amoxateT yvela, oRond cal-calke! 

masw.-i kvlav amowmebs, rom moswavleTa mier daxatuli 

samkuTxedebi daaxloebiT imxelebi iyos, ramxelebicaa 

naZvisebr nakvTze. 

aseve gairCeva Sedgenili marTkuTxedic. SeiZleba davinaxoT 

sami marTkuTxedi. 

Semdegi naxazis SekiTxva daismis ara marTkuTxedebis 

Sesaxeb (rac Zalian Znelia), aramed kvadratebis Sesaxeb. es 

araa Zneli: SesaZlebelia xuTi kvadratis danaxva. 

yvela danaxuli nakvTi cal-calke amoixateba, zomebis 

SeZlebisamebr SenarCunebiT. 

78/5/!`txsbgj!qbtvyfcj~/ SekiTxvebi Seexeba naswavl sivrciT 

mimarTebebsa da rigobiT ricxvebs _ mTw-s ydis suraTze.

- 202 - 

pirvel wreze moswavleebi naxuloben am suraTs da ise 

pasuxoben, meore wreze _ yvela Seinaxavs wignebs da 

warmodgeniT pasuxoben. 

SekiTxvaTa nimuSebi ix. 8.4-Si. 

g 68. kvgUfcbe!ebUwmb/!7.jt-!8.jt!ebTmb/!hbnfpsfcb/ 

79/2/! ojwUUb! hspwjt! ebxzpcb! kvgUfcbe! eb! bn! kvgUfcjt! 

ebUwmb/! `sbnefoj! pUyfvmjb@~! ! jhjwf! `! ebybuvm! tbhofc{f/! 

dafaze areulad mikrulia 18 wris grova. gamova moswavle da 

dafaze mikrul wreebis grovas daawyobs sameulebad. _ ramdeni 

sameuli miviReT? wreebi ki ar unda davTvaloT, sameulebi 

unda davTvaloT! ramdeni wrea _ ar gvainteresebs! ramdeni 

sameulia? {eqvsi}. 

masw.-i TiTiT saTiTaod Semoxazavs sameulebs: _ erTi 

sameuli, ori sameuli, sami sameuli, oTxi sameuli, xuTi 

sameuli da eqvsi sameuli! sul eqvsi cali sameulia. eqvsi 

cali sameuli! 

Sesruldeba kidev ori amdagvari savarjiSo, oRond sxva 

nivTebiT: 9 kubi daewyoba sameulebad (da bolos miviRebT: 

`sami cali sameuli~), 12 rgoli daewyoba eqvseulebad (òri 

cali eqvseulia~). 




marto davawyveT sxvanairad!} 

_ axla naxeT suraTi (68.1). risi grovebia? {samkuTxedebisa} 

rogoraa dawyobili samkuTxedebi? {oTxeulebad} ramdeni 

oTxeulia? {Svidi cali oTxeulia}. amasobaSi masw.-i 

dafaze daxatavs oTxkuTxedebs: sam xuTeuls. _ ra davxaze me 

dafaze? {oTxkuTxedebi} rogoraa isini dawyobili? 

{xuTeulebad} ramdeni xuTeulia? {sami cali xuTeulia}. 

79/3/!frwtjtb!eb!Twjejt!ebTmb!zpwfmhwbs!TftblsfcUb!kbnf. 

cbe/! yvelaferi keTdeba iseve, rogorc 66.2-Si, oRond eqvsisa 

da SvidisaTvis (mTw `6~ da `7~ da msr-s suraTi 68.2). 

79/4/!Ubwj!7/4.jt!Tfkbnfcb!eb!hbnfpsfcb/!moswavleebi cariel

- 203 - 

adgilas (68.3) daxazaven raime mravalkuTxeds (visac rogori 

moundebaT), gaaferadeben yviTlad an mwvaned, sazRvars lurjad 

Seaferadeben, wveroebs wiTlad moniSnaven da daTvlian. 

masw.-i, rogorc yovelTvis aseT dros, merxebs Soris dadis 

da muSaobas Tvalyurs adevnebs. roca daamTavreben, sxva 

sakiTxze gadava: _ gavixsenoT: radara nakvTebad iyofa 

mravalkuTxedebi? radara nakvTebad SeiZleba davajgufoT 

mravalkuTxedebi? {samkuTxedebi, oTxkuTxedebi, xuTkuTxedebi, 

eqvskuTxedebi, SvidkuTxedebi...}. 

gadavSliT mTw-s boloSi (geometriuli nakvTebi) da 

ganvixilavT mravalkuTxedebis nawils. masw.-i aCvenebs am 

nawils: _ ai, esaa dajgufebuli mravalkuTxedebi! radara 

nakvTebadaa dajgufebuli? yvela CamoTvaleT! {samkuTxedebi, 

oTxkuTxedebi, xuTkuTxedebi, eqvskuTxedebi, SvidkuTxedebi, 

rvakuTxedebi, cxrakuTxedebi, aTkuTxedebi da sxva mravalkuTxedebi} 

sul ramdeni jgufia? {cxra jgufia _ oTxkuTxedebi 

erT jgufad CavTvaloT, Tumca Sig marTkuTxedebi 

da kvadratebia calke gamoyofili} ra ferisaa xuTkuTxedTa 

jgufi? romeli jgufebia mwvane? (da ase Semdeg). 

79/5/!`lsjbybt!xjxjmfcj~/ imdagvarivea, rogorc 64.4, oRond 

nacvlad `Cik-Cik, gamoiCeka~-si iqneba `wiv-wiv, sabudaridan 

ezoSi gavida~. wiwilebi sabudaridan TandaTan gavlen ezoSi, 

mwvane mdeloze, sadac mziani Tbili amindia... mokle CanarTebs 

maswavle-beli Tavad moifiqrebs. arsebiTi ki isaa, rom xdeba 

raodenoba aTis daSla sxvadasxva SesakrebTa jamebad, Semdeg 

erTerTs akldeba erTi an ramdenime da es emateba meores. 

SekiTxvebia: _ ramdeni wiwilaa sabudarSi? ezoSi ramdenia? 

g 69. xzwjmfcbe!ebxzpcb/!hbnplm/!sjdywUb!ljcf{f/!ufyjmj/ 

7:/1/! 68.6-Si pasuxia: eqvsi cali xuTeuli. wignebi 

dawyobilia eqvs cal xuTeulad. 

7:/2/!xzwjmj!bov!psfvmj/!xzwjmfcbe!ebxzpcb!eb!bn!xzwjmUb! 

ebUwmb/!gakeTdeba imdagvarive savarjiSoebi, rogorebicaa 68.1- 

Si, oRond yoveli meore savarjiSo iqneba oreulTa

- 204 - 

(anu wyvilTa _ xan ase vityviT da xan _ ise) dawyobaze. 




marto davawyveT sxvanairad!}. 

bolos gairkveva, ra xatia da rogoraa dawyobili msr-s 

suraTze 69.1 {varskvlavebi dawyobilia aT cal wyvilad}. 

am sagakveTilo monakveTs bevri dro ar unda dasWirdes. 

7:/3/!psnprnfefcjboj!hbnplmfcb!sjdywUb!ljcf{f/! 

jer gavimeorebT ormoqmedebian Sekrebas (ix. 67.2). 

Semdeg yvelaferi _ iseve, rogorc 67.2-Si, oRond Sekrebis 

nacvlad iqneba gamokleba. 

am sagakveTilo monakveTs bevri dro ar unda dasWirdes. 

------------------------------------------------------------------------------------ 

JJJ.8/! Semogvaqvs axali cnebebi:! ufyjmj! yb{j! bov! ufyjmj-! 

Tflsvmj!eb!hbytojmj!ufyjmfcj

- 205 - 

gaferaddes _ nodars xom mTeli aTkuTxedis daxazva ar 

evaleba! masw.-i nodaris mier daxazul sazRvars carcs kidev 

erTxel gadaatarebs, raTa daxazuli monakveTebi (gverdebi) 

kargad gaasworos. 

_ ai, esaa mravalkuTxedis sazRvari! mas ewodeba texili 

xazi. texili. ai es rogori xazia? (daxazavs grZel monakveTs) 

{esaa monakveTi, swori xazi} diax, es xazi _ swori xazia, ai 

es ki _ TiTqos datexilia. datexiliviTaa da amitom ewodeba... 

ra? {texili}. ai me davxazav kidev erTs texils (daxazavs 

raime Sekrul texils, rva monakveTisagan Semdgars, ukve 

mravalkuTxedTan kavSiris gareSe). oTar, es ra nakvTia? 

{texili}. diax, texilia, texili xazi. risgan Sedgeba 

texili? {monakveTebisagan} es pirveli texili ramdeni 

monakveTisagan Sedgeba? {aTisagan} es meore texili? {rva 

monakveTisagan} diax, texili monakveTebisagan Sedgeba. axla me 

kidev or xazs davxazav. 

masw.-i daxazavs mesame, meoTxe, mexuTe da meeqvse xazebs: 

3) Sekrul mrud xazs; 4) axal Sekrul texil xazs; 

5) gaxsnil texil xazs; 6) Sekrul xazs, romelic 

nawilobriv kia texili, magram or adgilas gamrudebulia da 

momrgvalebul kuTxes qmnis. 

_ es mesame xazia. aris Tu ara igi texili? {ara} ratom? 

{monakveTebisagan ar Sedgeba} es meoTxe xazia. aris Tu ara igi 

texili? {ki} ratom? {sul monakveTebisagan Sedgeba} es mexuTe 

xazia. aris Tu ara igi texili? 

mosalodnelia pasuxTa sxvadasxvaoba: zogma SeiZleba Tqvas, 

rom araa texili, radgan dainaxvs, rom es xazi araa Sekruli 

(Tumca terminebs `Sekruli~ da `gaxsnili~ am gakveTilze ar 

vaxsenebT). masw.-i ityvis: _ esec texilia, radganac sul 

monakveTebisagan Sedgeba. es ki meramdene xazia? {meeqvse xazia} 

aris Tu ara igi texili? 

TandaTan gairkveva, rom ara, radganac sul (am sityvis 

gamokveTiT) monakveTebisagan ar Sedgeba. texili mxolod (am 

sityvis gamokveTiT) monakveTebisagan unda Sedgebodes, mxolod

- 206 - 

swori xazebisagan, es xazi ki ama da am adgilebSi araa swori! 

_ aqvs Tu ara am xazs monakveTebi? {ki aqvs sami monakveTi} 

magram mainc araa texili, radgan... ra? {radgan yvelgan ara 

aqvs monakveTebi}. 

_ axla vin ityvis kargad: rogor xazs hqvia texili? 

rogori xazia texili? {texili iseTi xazia, romelic 

mxolod monakveTebisagan Sedgeba; texili ewodeba iseT xazs, 

romelic mxolod monakveTebisagan Sedgeba}. 

_ axla daakvirdiT suraTs (69.3). ramdeni xazia aq 

daxazuli? {cxra} romelia maTgan texili? pirveli Tu aris? 

{ki} ratom? {mxolod monakveTebisagan Sedgeba} meore? {ki}... 

(da ase Semdeg) meeqvse? {ara} ratom? {zemodan ara aqvs 

monakveTi, momrgvalebulia, texili ki momrgvalebuli ar 

SeiZleba iyos!} meSvide xazi Tu aris texili? {aris} ratom? 

{mxolod monakveTebisagan Sedgeba} merve? {ara} ratom? {ori 

xazisgan Sedgeba, erTi monakveTia da erTi _ ara} mecxre? 

{texilia} ratom? {mxolod monakveTebisagan Sedgeba}. maS, 

ramdeni xazia aq sul? {cxra} ramdeni yofila maTgan texili? 

{Svidi} ramdeni ar yofila? {ori} romeli da romeli? 

{meeqvse da merve}. 

7:/5/!`nbUfnbujlvsj!tjuzwfcjt!ebtbyfmfcb~/ ix. 38.4. cxadia, 

sityvebs emateba mas Semdeg naswavli yvela maTematikuri 

sityva: Sesakrebi, jami, sxvaoba, saklebi, maklebi, 

mravalkuTxedi, sazRvari, wvero, texili, SesaZlebloba, 

Tanmimdevroba da ase Semdeg. 

g 70. kvgU/!ebxzpcb/!9.jt-!:.jt!ebTmb/!ufyjmjt!ebyb{wb/ 

81/2/! xjobtxbs! npdfnvm! kvgUfcbe! ebxzpcb;! èbbxzwf! )bo;! 

ebybuf*!pUyj!dbmj!tbnfvmj"!)bo;!yvUj!dbmj!xzwjmj*~! 

gakeTdeba imdagvarive savarjiSoebi, rogorebic iyo 69.1-Si, 

oRond erTi siaxliT: jufTebi winaswar araa dawyobili, maT 

Tavad moswavle awyobs an xatavs. magaliTad, moZravi wreebi 

Tu kvadratebi erTad yria kolofSi, Tanac ufro meti 

raodenobiT, vidre jufTebis dasawyobadaa saWiro. moswavle

- 207 - 

TviTon amoiRebs maT da awyobs `sam cal eqvseulad~ Tu `rva 

cal wyvilad~. gakeTdeba oTxi amdagvari savarjiSo (or-ori 

gakeTdeba erTdroulad sxvadasxva moswavleTa mier). Semdeg 

gakeTdeba kidev oTxi (or-ori) savarjiSo dafaze carciT 

daxatvaze: _ Sen daxate jvrebi _ cxra cali oTxeuli, Sen ki 

daxate mravalkuTxedebi _ ori cali sameuli. Sen daxate 

texilebi _ ori cali oTxeuli, Sen ki _ swori xazebi _ 

xuTi cali rvaeuli. 

bolos, msr-s cariel adgilas (70.1) klasi daxatavs 

mraval-kuTxedebs _ sam cal oreuls (wyvils). 

81/3/!swjtb!eb!dysjt!ebTmb!zpwfmhwbs!TftblsfcUb!kbnfcbe/! 

keTdeba igive, rac 66.2-Si, oRond rvisa da cxrisaTvis 

(mTw-s `8~ da `9~ da msr-s suraTi 70.2). 

81/4/!ufyjmjt!cpmpfcj!eb!xwfspfcj/!ufyjmjt!ebyb{wb/! 

_ gaixseneT: rogori xazia mravalkuTxedis sazRvari? {texili 

xazi} ra aris texili? {xazi, romelic Sedgeba mxolod 

monakveTebisagan, swori xazebisagan} maS risgan Sedgeba 

texili? {mxolod monakveTebisagan, swori xazebisagan, sxva 

araferi ar unda hqondes!}. 

dafaze daixazeba erTi daxrili monakveTi, erTi gaxsnili 

texili da erTi Sekruli texili. _ ra aris es marcxena 

nakvTi? {mo-nakveTi} gamovides ana da moniSnos am monakveTis 

boloebi. manamde ki nanam gviTxras _ ramdeni bolo aqvs 

monakveTs? {ori} yvela monakveTs ori bolo aqvs? {diax} ai, 

anam moniSna monakveTis boloebi. ra aris monakveTis bolo _ 

xazi, wre, rgoli, kvadrati Tu ra? {monakveTis bolo aris 

wertili} diax, Cven xom viciT, rom nakvTis wvero aris 

wertili, aseve monakveTis boloc aris wertili. 

_ axla, ra aris es Sua nakvTi? {texili} gamovides vano 

da moniSnos am texilis bolo wertilebi. ramdeni bolo aqvs 

am texils? {ori} axla oTari gamovides da moniSnos am 

texilis wveroebi! ramdeni wvero hqonia am texils? {sami} 

diax, ai erTi, ori da sami! erTi bolo, meore bolo, erTi 

wvero, meore wvero da mesame wvero.

- 208 - 

_ ra aris es marcxena nakvTi? {texili} aqvs Tu ara mas 

bolo wertilebi? {ara, ara aqvs} ese igi, yvela texils aqvs 

Tu ara bolo wertilebi? {ara} yvela monakveTs aqvs Tu ara 

bolo wertilebi? {ki} zog texils Tu aqvs? {ki} wveroebi 

Tu aqvs yvela texils? {ki, wveroebi yvela texils aqvs, 

boloebi yvelas ara aqvs, magram wveroebi yvelas aqvs}. diax, 

wveroebi yvela texils aqvs da texilis dasaxazvad jer misi 

wveroebi unda movniSnoT, Semdeg ki isini SevaerToT... riTi? 

riTi unda SevaerToT dasaxazi texilis wveroebi? 

{monakveTebiT} diax, jer texilis wveroebi unda movniSnoT, 

Semdeg ki isini SevaerToT monakveTebiT. 

daxazaven msr-Si texilebs (70.3). 

81/5/!`lsjbybt!xjxjmfcj~/ imdagvarivea, rogorc 64.4, oRond 

nacvlad `Cik-Cik, gamoiCeka~-si iqneba `wiv-wiv, ezodan gareT 

gaZvra~. zogierTi wiwila RobeSi rom RriWoebia, iqidan gaZvreba 

gareT. magram gareT rom marto aRmoCndeba, maleve ukan 

SemoZvreba. gareT agreTve ramisa SeeSindeba (ZaRlis, urmis, 

manqanisa...) da saCqarod miirbens deda kriaxasTan. mokle 

CanarTebs masw.-i Tavad moifiqrebs. arsebiTi ki isaa, rom xdeba 

raodenoba aTis daSla sxvadasxva SesakrebTa jamebad, Semdeg 

erTerTs akldeba erTi an ramdenime da es emateba meores. 

garda amisa, emateba erTi sakiTxic. yuradReba maxvildeba 

`Sig \ gareT~ mimarTebaze. SekiTxvebia: _ ramdeni wiwilaa 

ezoSi? ezos gareT? _ magram ramdenjerme isec moxdeba, rom 

mousvenari wiaqa (an zogjer masTan erTad sxvac, romelic 

wiaqas ahyveba celqobaSi) Robeze Sefrindeba. am SemTxvevaSi 

daemateba SekiTxva: _ ezoSia xuTi, ezos gareTaa sami, magram 

xuTi da sami ramdenia? {rva} aTa-mde ramdeni aklia? {ori} es 

ori wiwila sadRaa _ ezoSia Tu ez-os gareT? {sazRvarzea, 

Robezea!} diax, xuTi SigniTaa, sami gareTaa da ori _ 

sazRvarzea. xuTi, sami da ori _ sul ramdenia? {aTi}. 

81/6/ _ bavSvebo, axla Cven unda daviwyoT cifrebis wera. 

sul ramdeni cifri viswavleT? CamoTvaleT yvela! {nuli, 

erTi, ... cxra _ aTi ki cifri araa!}. dRes viswavliT erTianis

- 209 - 

weras. es sul advilia. 

masw.-i dafaze Zalian didad dawers erTians. Tan yuradRebas 

amaxvilebs mis nawilebze da xelis moZraobaze. moswavleebi 

msr-Si iwyeben erTianebis gamoweras. masw.-i mxolod imas 

aqcevs yuradRebas, rom cifris werisas moswavlis xelis 

moZraobis mimarTuleba iyos marTebuli. 

70.5-is erT striqons klasSi Seavseben, meores _ Sin. 

g 71. `zpwfmj~-!ÙwjUfvmj~/!Tflsvmj-!hbytojmj!ufyjmfcj/ 

82/2/! ufsnjofcj! `zpwfmj~! eb!ÙwjUfvmj~! `! kvgUfcUbo! 

eblbwTjsfcjU/! keTdeba imdagvarive savarjiSoebi, rac 70.1-Si, 

oRond erTi damatebiT: jufTebis dawyobis Semdeg masw.-i 

ikiTxavs: _ am grovaSi ramdeni kvadratia? {rva} amaSi? {rva} 

amaSi? {amaSic rva} diax, TviTeul am grovaSi _ rva kvadratia. 

amaSic rvaa, amaSic, amaSic (Tan TiTs daadebs). 

yovel grovaSi rva kvadratia. 

moswavleebsac aTqmevinebs sityvebs `yoveli~ da `TviTeuli~ 

(am SemTxvevaSi isini sinonimebia). 

82/3/!psnprnfefcjboj!Tflsfcb.hbnplmfcb!sjdywUb!ljcf{f/! 

jer ricxvTa kibeze ramdenimejer Sesruldeba naswavli 

ormoqmedebiani gamoTvlebi: Sekreba da gamokleba _ imdagvarive 

savarjiSoebi, rac 67.2 da 69.2-Si. 

Semdeg sruldeba ramdenime axali gamoTvla: 

8 _ 5 + 2, 6 + 3 _ 5 da sxva. 

erTi moswavle moqmedebas did saklaso ricxviT kibeze 

asrulebs, sxvebi _ Tav-TavianT mTw-Si. 

82/4/!Tflsvmj!eb!hbytojmj!ufyjmfcj/! 

dafaze daixazeba ori texili _ erTi Sekruli da meore _ 

gaxsnili. _ romel texils aqvs boloebi? {marjvenas} romelia 

amaTgan Sekruli texili? {marcxena} gaxsnili texili? 

{marjvena} aba cotnem gviCvenos, ratomaa marjvena texili 

gaxsnili! (cotne TiTs daadebs im adgils, sadac texili 

gaxsnilia). diax. warmoidgineT, rom wertili am texilis 

SigniTaa (masw.-i dasvams wertils Sekruli texilis SigniT). es

- 210 - 

wertili daWerilia, mas gareT xom ver gamovasrialebT! 

ratom? {imitom, rom texili Sekrulia, arsad araa gaxsnili 

da wertils ver gamovasrialebT} aqedan? (masw.-i dasvams 

wertils gaxsnili texilis or monakveTs Soris). aqedan Tu 

SeiZleba wertilis gamosrialeba? {ki} rogor? cotnem 

gviCvenos (cotne maCvenebel joxs daadebs wertils da aCvenebs 

gzas, rogor SeiZleba misi `gamosrialeba~). gaxsnili texiliT 

wertilis daWera SeiZleba? {ara} Sekruli texiliT? {ki}. 

_ daakvirdiT naxazs (71.3). ramdeni striqonia texilebisa? 

{ori} TiTo striqonSi ramdeni texilia? {xuTi} sul ramdeni 

texilia? {aTi} texilebis ramdeni wyvilia aq daxazuli? 

{xuTi cali wyvili} diax. axla vin mixvdeba, ra 

kanonzomierebiTaa daxazuli es texilebi, rogorebia zeda da 

qveda striqonebi? {zeda striqonSi sul Sekruli texilebia, 

qvedaSi _ sul gaxsnilebi} hgavs Tu ara erTmaneTs erT 

svetSi daxazuli texilebi? {Zalian hgavs} magram maT Soris 

ra gansxvavebaa? {zeda Sekrulia, qveda _ gaxsnili}. axla 

yvela texilis yvela bolo wertili moniSneT wiTlad! 

zemoTa striqonSi Tu moiniSna rame? {ara} ratom? {imitom, 

rom iq mxolod Sekruli texilebia, maT ki boloebi ara aqvs} 

axla yvela Sekruli texilis yvela wvero moniSneT mwvaned, 

yvela gaxsnili texilis yvela wvero ki moniSneT lurjad! 

82/5/! ùzvjmjb! Uv! nbsUbmj~! _ ix. 61.4. winadadebaTa 

nimuSebi: 1) yovel monakveTs ori bolo aqvs. 

2) yovel texils ori bolo aqvs. 

3) yovel texils ori wvero aqvs. 

4) yovel monakveTs ori wvero aqvs. 

5) yovel texils xuTi wvero aqvs. 

6) zog texils xuTi wvero aqvs. 

7) yoveli texili mxolod monakveTebisagan Sedgeba. 

8) yoveli texili ori monakveTisagan Sedgeba. 

9) texili swori xazia. 

10) monakveTi swori xazia. 

11) arcerT texils ara aqvs bolo wertilebi.

- 211 - 

12) zog texils ara aqvs bolo wertilebi. 

13) zog texils ara aqvs wveroebi. 

14) zog monakveTs aqvs wveroebi. 

15) gaxsnil texils wveroebi ara aqvs. 

16) mravalkuTxedis sazRvari texili xazia. 

17) yvelanairi texili mravalkuTxedis sazRvaria. 

18) kvadrati mravalkuTxedia. 

19) yvelaniari samkuTxedi mravalkuTxedia. 

20) yvelaniari mravalkuTxedi samkuTxedia. 

21) wre mravalkuTxedia. 

22) mravalkuTxedi rgolia. 

23) yvelanairi marTkuTxedi mravalkuTxedia. 

24) yvelanairi mravalkuTxedi kvadratia. 

25) yvelanairi mravalkuTxedi marTkuTxedia. 

26) mravalkuTxedis sazRvari Sekruli texilia. 

27) mravalkuTxedis sazRvari gaxsnili texilia. 

28) Sekrul texils boloebi ara aqvs. 

29) gaxsnil texils boloebi ara aqvs. 

30) Sekrul texils wveroebi ara aqvs. 

g 72. ebUwmb!kvgU/!Tfls/.hbnplm/!{fqjsbe/!ufyjm/!ebyb{wb/ 

83/2/!UwbmtbDjope!npdfnvmj!sbpefopcjt!ebUwmb!kvgUfcbe; 

`daTvale oTxeulebad! daTvale wyvilebad! daTvale 

aTeulebad!~ saWiro pasuxis nimuSi: èqvsi cali oTxeuli~. 

keTdeba imdagvarive savarjiSoebi, rac 70.1-Si, oRond erTi 

damatebiT: jufTebis dawyobis Semdeg masw.-i moswavles 

daavalebs: _ axla es varskvlavebi daTvale oTxeulebad! 

varskvlavebi ki ar daTvalo, oTxeulebad unda daTvalo! 

Tuki moswavleebs gauWirdebaT, masw.-i ukarnaxebs: _ unda 

daTvalo oTxeulebad, unda daTvalo, sul ramdeni oTxeulia! 

saWiro pasuxis nimuSia: èqvsi cali oTxeuli~. 

yoveli dawyobis Semdeg daismis amdagvari sakiTxi: daTvale 

wyvilebad! daTvale aTeulebad! (da sxva). 

masw.-i drodadro moswavleebs aTqmevinebs sityvebs `yoveli~ 

da `TviTeuli~ (am SemTxvevaSi isini sinonimebia), iseve,

- 212 - 

rogorc 71.1-Si. 

83/3/!bewjmj!psnprnfefcjboj!Tflsfcb.hbnplmfcb!{fqjsbe/! 

savarjiSoTa nimuSebi: 

2 + 4 + 4; 8 _ 3 _ 5; 3 + 6 _ 7; 7 _ 4 + 3. 

sasurvelia, isini zepirad Sesruldes. Tuki moswavles 

gauWirdeba, moiSvelios ricxvTa kibe mTw-Si. 

83/4/!Tflsvmj!eb!hbytojmj!ufyjmfcjt!ebyb{wb/! 

dafaze daismeba wertilebi da saklaso saxazavis gamoyenebiT 

daixazeba TiTo-TiTo Sekruli da gaxsnili texili. 

yuradReba gamaxvildeba imaze, rom gaxsnili texilis xazvas 

Cveulebrivad vamTavrebT, Sekruli texilis dasaxazavad ki 

unda davimaxsovroT, Tu romeli wertilidan daviwyeT xazva 

da bolos isev im wertilSi unda davasruloT. saidanac 

daviwyeT, iqve unda davamTavroT! _ ratom? {imitom, rom xazi 

unda Seikras}. 

msr-Si (72.3) moniSnuli wertilebis mixedviT daxazaven 

gaxsnil, Sekrul, gaxsnil texilebs. Semdeg TviTon moniSnaven 

wertilebs da daxazaven kidev TiTo-TiTo gaxsnil da Sekrul 

texilebs. 

83/5/ `lsjbybt! xjxjmfcj~/ imdagvarivea, rogorc 70.4. 

arsebiTad yvelaferi isevea, oRond cota sxva rameebic xdeba: 

moiRrubleba da wvima wamova, rac wiwilebs Zalian SeaSinebs 

da sabudarSi Secvivdebian, diasaxlisi sakenks dauyris qaTmebs 

da kriaxa iqiT gaiqceva da sxva. mokle-mokle CanarTebs Tavad 

masw.-i moifiqrebs. 

arsebiTi ki igivea: aTis daSla SesakrebTa jamebad, 

ramdenimes mimateba Tu mokleba, mimarTeba `SigniT \ gareT \ 

sazRvarze~. 

g 73. U/!:/2.jt!hbnfpsfcb/!`.jU!nfupcb~!tbzpgbdy/!bnpdbofcTj/ 

84/2/!Ubwj!:/2.jt!Tfkbnfcb!eb!hbnfpsfcb/! 

keTdeba imdagvarive savarjiSoebi, rac 72.1-Si. 

masw.-i amasobaSi dafaze daxatavs 18 monakveTisagan Semdgar 

areul grovas. _ es monakveTebi unda daiwyos sameulebad. raki

- 213 - 

daxatul monakveTebs ver aviRebT, ver avamoZravebT, ra vqnaT? 

{isini ise unda SemovxazoT, rom sul sameulebi miviRoT} 

gamovides nukri da Semoxazos (sasurvelia, feradi carciT)! 

sosom ki daTvalos sameulebad! {eqvsi cali sameuli}. 

Semdeg suraTze (73.1) borjRalebi unda daiwyos 

oTxeulebad. isini ise unda Semoixazos, rom sul oTxeulebi 

miviRoT. Semoxazaven. _ borjRalebi daTvaleT oTxeulebad! 

{eqvsi cali oTxeuli}. 

84/3/! `.jU! nfupcb~! fsUnprnfefcjbo! tbzpgbdypwsfcp! bnpdb. 

ofcTj/! ! jer suraTis (73.2) mixedviT swrafad gavameorebinebT 

`-iT metobas~. _ risi da risi grovaa aq daxatuli? (ruxi da 

vardisferi wreebisa). romelia meti? {vardisfrebi} ramdeniT? 

{samiT} saidan ici? {wreebi dawyvilebulia, sami vardisferi 

wre ki uwyvilodaa darCenili} vardisferi wreebi samiT metia, 

vidre ruxebi. 

amis Semdeg amoixsneba saTanado sayofacxovrebo amocanebi. 

rogorc wesi, amocanebi safexurebrivad amoixsneba _ ix. 

saerTo miTiTeba II-4 (25.2-is win). aq pirvel safexurebs 

(pirobis gameoreba) aRar vaxsenebT _ TavisTavad vgulisxmobT. 

`-iT met-nakleboba~, maT Soris sayofacxovrebo amocanebSi, 

am droisaTvis imdenad kargadaa damuSavebuli, rom saTanado 

erTmoqmedebiani amocanebis amoxsna Zalian advili unda iyos. 

Tuki romelime moswavles mainc gauWirdeba, mxolod maSin 

daixatos dafaze sqema: ai viTom es xazebi _ mWedlis 

Svilebia... ise ki sqemis daxatva aRaraa saWiro. 

_ tyeSi cxovroben megobrebi _ ciyunia da baWia. baWiam iSova 

eqvsi kakali, ciyuniam ki iSova amdenive kakali da kidev 

ori kakali. romels aqvs meti kakali da ramdeniT? saidan ici? 

_ mWedels ori Svili hyavs, mecxvares _ samiT meti. 

ramdeni Svili hyavs mecxvares? 

_ naTiam oTxi nakvTi gaaferada, cicinom _ samiT meti. 

ramdeni nakvTi gauferadebia cicinos? 

84/4/! zwfmb! Tftb[mfcmpcjt! bnpxvswjU;! sbnefoj! xsf! bo! 

shpmj! Tfj[mfcb! ebwjobypU! obyb{{f! )bsbvnfuft! frwtjtb*/! nbUj!

- 214 - 

dbm.dbmlf! bnpybuwb/! ganvixilavT suraTs (73.3). masze 

SeiZleba davinaxoT sami wre, erTmaneTSi Calagebulni. 

rgolebi ufro rTuladaa. SeiZleba davinaxoT sami rgoli. 

yvela cal-calke amoixazeba da saTanadod gaferaddeba. 

84/5/!`lsjbybt!xjxjmfcj~/ imdagvarivea, rogorc 72.4. oRond 

cota sxva rameebi xdeba _ mokle-mokle CanarTebs Tavad masw.-i 

moifiqrebs. 

arsebiTi ki igivea: aTis daSla SeskrebTa jamebad, ramdenimes 

mimateba Tu mokleba, mimarTeba `SigniT \ gareT \ sazRvarze~. 

!g 74. kvgUfcjt!Dbxfsb/!npoblwfUfcjt!ebobywb!obyb{{f/! 

85/2/!ebxzpcjmj!kvgUfcjt!Dbxfsb;!4 · 5/!xbljUywb;!`tbnj!dbmj! 

pUyfvmj~/! masw.-i dafasTan gamoiyvans moswavles da daawyobinebs 

samkuTxedebis sam cal oTxuls. _ ramdeni nakvTia 

TviTeul am grovaSi? {oTxi} sul ramdeni oTxeulia? {sami} 

hoda es ase Caiwereba: 3 · 4. ras gviCvenebs es samiani (masw.-i 

3-ians daadebs TiTs)? {rom sami calia oTxeuli} es oTxiani? 

(masw.-i 4-ians daadebs TiTs)? {rom oTxeulebia, rom TiTo 

grovaSi oTxia}. es wertili raRas gviCvenebs? {rom calia, 

wertili calis niSania!} maS, ase CavwereT (daadebs TiTs 

gamosaxulebas). wavikiTxoT es: sami (miuTiTebs 3-ians) cali 

(miuTiTebs · -s) oTxeuli (miuTiTebs 4-ians). 

gamoiyvans sxvas da daaxatvinebs monakveTebis rva cal 

wyvils. _ uCa, daTvale wyvilebad! {rva cali wyvili} rogor 

CavweroT? jer ra cifri unda davweroT? {rviani} diax, ai, 

davwere rviani (dawers masw.-i). axla ra davwero? {wertili, 

cali} ai, davwere. axla? {oriani}. ratom? {imitom, rom 

wyvili _ igive oreulia}. 

Semdeg masw.-i dafaze dawers gamosaxulebebs: 

9 · 4, 4 · 5, 6 · 6. moswavleebi adgilidan waikiTxaven maT. 

suraTze (74.1) TviTeul CarCos SeaerTeben Sesabamis CanawerTan 

qveda striqonidan. Tan yvela am Canawers waikiTxaven. 

85/3/! `.jU! oblmfcpcb~! fsUnprnfefcjbo! tbzpgbdypwsfcp! 

bnpdbofcTj/ isevea, rogorc 73.2.

- 215 - 

jer daakvirdnen naWrelas suraTze (74.2) da dauTvlelad, 

swrafad gaarkvion, romelia meti da ramdeniT _ vardisferi Tu 

ruxi samkuTxedebi. amisaTvis unda warmoidginon dawyvileba (an 

marTla SeaerTon mokle-mokle daxrili xazebiT). aRmoCndeba, 

rom boloSi morCeba erTi dauwyvilebeli ruxi samkuTxedi. 

_ baWiam iSova eqvsi kakali, ciyuniam ki iSova amdenive 

kakali, magram ori kakali maSinve SeWama. romels aqvs meti 

kakali da ramdeniT? saidan ici? 

_ baWiam ipova xuTi nuSi, ciyuniam ki ipova samiT meti 

nuSi. ramdeni nuSi upovia ciyunias? {cxra} rogor gaige? 

{eqvss mivumate sami} ratom miumate, iqneb, unda gamogeklo? 

{mivumate imitom, rom ciyunias upovia samiT meti nuSi}. 

_ baWiam ipova rva wabli, ciyuniam ki ipova samiT naklebi 

wabli. ramdeni wabli upovia ciyunias? {xuTi} rogor gaige? 

{rvas gamovakeli sami} ratom gamoakeli, iqneb, unda 

migematebina? {gamovakeli imitom, rom ciyunias upovia samiT 

naklebi nuSi}. 

_ saqaTmeSi 9 dedalia. mamlebis raodenoba ki SvidiT 

naklebia. ramdeni mamalia saqaTmeSi? 

_ xvalisTvis Tqven TviTon moigoneT axali amocana. gogonebma 

moigonon amocana oTxiT metobaze, biWebma _ oTxiT naklebobaze. 

85/4/!zwfmb!Tftb[mfcmpcjt!bnpxvswjU;!sbnefoj!npoblwfUj!bo! 

ufyjmj! Tfj[mfcb! ebwjobypU! obyb{{f! )bsbvnfuft! frwtjtb*/! nbUj! 

dbm.dbmlf! bnpybuwb/! ganvixilavT naxazs (74.3). _ ramdeni 

texilia daxazuli? {ori} rogori texilebia? {erTi Sekruli, 

erTi _ gaxsnili} ramdeni monakveTis danaxvaa SesaZlebeli 

Sekrul texilSi? {samis} amoxazeT cal-calke! axla isev am 

Sekrul texils daakvirdiT. ramdeni texilis danaxvaa masSi 

SesaZlebeli? ar gamogrCeT Camalulebi! 

sul oTxi texilis danaxvaa SesaZlebeli: erTia TviTon 

Sekruli texili, meore, mesame da meoTxe ki miiReba or-ori 

mezobeli monakveTis gaerTianebiT. dafazec daixazeba, 

moswavleebi yvela am texils cal-calke amoxazaven. 

_ axla davakvirdeT qveda texils. rogori texilia?

- 216 - 

{gaxsnili} ramdeni monakveTisagan Sedgeba igi? {samisagan} 

amovxazoT samive cal-calke! kargi, axla amovxazoT yvela 

Camaluli texili! (sul oria: pirveli+meore monakveTi da 

meore+mesame monakveTi). 

85/5/!èbwehfU!Dbobxfsjt!njyfewjU~/ masw.-i dafaze dawers: 

5 · 3. moswavleebi unda gamovidnen TiTo-TiTod, Tanac zustad 

im TanmimdevrobiT, rogorc sxedan: jer marcxena rigis 

pirveli merxis marcxena, Semdeg misi mezobeli, Semdeg meore 

merxi da ase Semdeg. SeiZleba am Tanmimdevrobis Secvla, 

mTavari isaa, rom klasma kargad icodes, ra TanmimdevrobiT 

unda gamovidnen da es Tanmimdevroba aRar unda Seicvalos arc 

am gakveTilze da arc SemdgomSi. 

moswavleebi gamodian da dgebian ise, rom miviRoT 

moswavleTa 5 · 3. 

Semdeg masw.-i sxva Canawers dawers, gamodian is 

moswavleebi, visac pirvel jerze ar mouwia gamosvlam (oRond 

kvlav imave TanmimdevrobiT) da ase Semdeg. 

g 75. kvgUfcjt!ebxzpcb/!Ubwj!9/3.jt!eb!8/4.jt!hbnfpsfcb/ 

86/1/ ramdenime gogona amoxsnis biWebis mier Sedgenil 

amocanas, ramdenime biWi _ gogonebis mier Sedgenils. 

74.8-Si unda amoexazaT sami monakveTi da ori Camaluli 

texili: pirveli+meore monakveTi da meore+mesame monakveTi. 

86/2/!kvgUfcjt!ebxzpcb!xjobtxbs!npdfnvmj!Dbobxfsjt!njyfe. 

wjU/ jer swrafad gavameorebinebT 74.1-s. Semdeg masw.-i dafaze 

dawers: 6 · 5, gamoiyvans moswavles da daavalebs: _ daawyve 

amdeni raodenobis Cxiri! sanam moswavle daawyobs, masw.-i dawers 

2 · 9 da sxva moswavles daavalebs amdeni raodenobis mar- 

TkuTxedis dawyobas dafaze. Semdeg dawers 4 · 4 da sxva moswavles 

daavalebs amdeni raodenobis Sekruli texilis dafaze 

daxatvas. Semdeg dawers 5 · 3 da sxva moswavles daavalebs 

amdeni raodenobis gaxsnili texilis dafaze daxatvas. 

moswavlem davalebis Sesrulebisas xmamaRla unda Tqvas, 

rasac akeTebs: _ unda davawyo Cxirebis eqvsi cali xuTeuli,

- 217 - 

ai erTi xuTeuli, meore xuTeuli... 

Semdeg klasi msr-Si, cariel CarCoSi 75.1 Caxatavs 7 · 4 

raodenobis texilebs. 

86/3/!Ubwj!9/3.jt!hbnfpsfcb!eb!Tfkbnfcb/ 

damuSavdeba ukve naswavlis magvari amocanebi: 

1) naTias xuTi TeTri aqvs, giorgis _ xuTiT meti TeTri. 

ramdeni TeTri aqvs giorgis? 

2) TinaTini Svidi wlisaa, misi Zma _ cxrisa. romelia maT 

Soris umcrosi da ramdeni wliT? 

3) klasSi 8 gogonaa. vaJebi 2-iT metni arian. ramdeni vaJia 

klasSi? 

4) vaJas unda gaeferadebina sul rva nakvTi. man moaswro 

xuTi nakvTis gaferadeba. ramdeni nakvTi darCa gasaferadebeli 

vaJas? 

5) Sav dedaRors 7 goWi hyavs, TeTr dedaRors _ sami 

goWiT naklebi. ramdeni goWi hyavs TeTr dedaRors? 

86/4/! Ubwj! 8/4.jt! Tfkbnfcb! eb! hbnfpsfcb/! ganvixilavT naxazs 

75.3. _ ramdeni Sekruli texilis danaxvaa SesaZlebeli am 

naxazze? ar gamogrCeT Camalulebi! {samisa} amoxateT calcalke, 

oRond gverdiT, marjvniv! (Tan dafazec daxatavs is 

moswavle, romelic sxvebze adre daasrulebs rveulSi xatvas). 

ramdeni monakveTis danaxvaa SesaZlebeli am naxazze? ar 

gamogrCeT Camalulebi! {cxrisa} amoxateT cal-calke, isinic 

gverdiT, marjvniv, qveda adgili sxva ramisaTvis dagvWirdeba! 

gaxsnili texilebis amowurva Znelia. sul maTi raodenoba 

Zalian didia. masw.-i ityvis amas da daiwyebs TamaSis saxiT: 

86/5/ `wjo!bnpyb{bwt!zwfmb{f!nfu!hbytojm!ufyjmt@~ 

masw.-i dainiSnavs dros (daaxloebiT 4 wuTi). Semdeg 

brZanebs: _ yvelam dados fanqari! da gamoavlens ramdenime 

gamarjvebuls. 

TfojTwob; 75.3-is magvar naxazebze sinamdvileSi ufro meti 

Camaluli texilebis danaxvaa SesaZlebeli. kerZod, mTeli 

naxazi SeiZleba ganxilul iqnas erT xazad. magram es aris 

rTuli, TavisTavis gadamkveTi xazi. amgvar xazebs ki pirvel

klasSi ar ganvixilavT. 

- 218 - 

g 76. ebUwmb!fsUfvmfcbe/!xsjt!tb{Swbsj/ 

87/2/!ebUwmb!fsUfvmfcbe;!!8 · 1, 5 · 1, 9 · 1!!eb!tywb/ 

jer swrafad gavameorebinebT 75.1-s. Semdeg masw.-i Zlier 

moswavles daavalebs: _ gamodi da daawyve kubebi, rva cali 

erTeuli! 

Tan dafaze dawers 8 · 1. Semdeg dawers 5 · 1 da sxvas daavalebs 

amdeni raodenobis marTkuTxedebis daxazvas. Semdeg 

klasi msr-Si (76.1) TviTeul CarCos xaziT SeaerTebs Sesabamis 

CanawerTan (gamosaxulebasTan) qveda striqonSi. 

_ maS, ramdeni yofila cxra cali erTeuli? {es yofila 

imdenive, ramdenicaa cxra, cxra cali erTeuli igivea, rac 

cxra}. 

------------------------------------------------------------------------------------ 

II-9/! kbnjtb! eb! tywbpcjt! Tfebsfcbt! sjdywUbo-! nbU! Tpsjt! 

tbzpgbdypwsfcp! bnpdbofcTj/! sbnefoj! blmjb! sjdywt! nfpsf! 

sjdywbnef!eb!bnbt!wblbwTjsfcU!tywbpcbtUbo (viTardeba wina Tavis 

Tema, oRond Tan kavSirdeba mocemuli ori ricxvis sxvaobasTan). 

masw.-s ar unda egonos, TiTqos bavSvisTvis es TavisTavad cxadia: 

rom erTi ricxvi meoreze metia maTive sxvaobiT. 

------------------------------------------------------------------------------------ 

87/3/!`.jU!nfupcjt~!eblbwTjsfcb!sjdywfcjt!tywbpcbtUbo/! 

dafaze magnituri wreebis `dominoseburi~ wiTeli aTeuli 

da TeTri eqvseulia. _ romelia amaTgan meti? {aTeuli, 

wiTlebi} modi, dawyvilebiT gavigoT, Tu ramdeniTaa meti aTi, 

vidre eqvsi (gamova moswavle da Cveulebriv daawyvilebs). 

ramdeniT meti yofila? {oTxiT} saidan ici? {oTxi wiTeli 

mogvrCa} (masw.-i iqve kidev erT wiTel aTeuls daawyobs). axla 

aTs gamoakeli eqvsi! (moswavle aiRebs eqvs rgols). risi 

tolia sxvaoba? {oTxisa} aTi ramdeniTaa meti, vidre eqvsi? 

{oTxiT} diax, aTi imdeniTaa eqvsze meti, ramdenicaa aTisa da 

eqvsis... ra? {sxvaoba} diax, sxvaoba gviCvenebs, Tu didi ricxvi 

ramdeniTaa meti, vidre patara! 

Semdeg moswavleebi gadaSlian mTw-s oceulis

- 219 - 

TvalsaCinoebaze (gv. 17). cxadia, ricxvebi 10, 11, ..., 19 _ jer 

ar gvWirdeba. _ sul ramdeni svetia aq? {aTi svetia}. yovel 

(am sityvis gamokveTiT) svetSi erTidaimave raodenobisaa 

wiTeli wreebi. ramdenia? {aT-aTi} bakur (susti moswavle), 

mecxre svetSi ramdeni iqneba wiTeli wre? {aTi} saidan ici? 

{yovel wreSi xom aTia, amitom mecxreSic aTi iqneba}. mayvala 

(yvelaze susti moswavle), meoTxe svetSi ramdeni iqneba 

wiTeli wre? {aTi} saidan ici? {yovel wreSi xom aTia, amitom 

mecxreSic aTi iqneba}. mwvane wreebi Tua erTidaimave 

raodenobiT? {ara}. aba, kanonzomierebas vin aRmoaCens? {pirvel 

svetSia erTi cali mwvane wre, meoreSi _ ori, mesameSi _ sami, 

da ase imatebs erTiT!). 

_ kargi, axla moZebneT meSvide sveti, daakvirdiT. ramdeni 

wiTeli wrea masSi? {aTi} mwvane wre ramdenia? {Svidi}. aTi 

ramdeniTaa meti, vidre Svidi? {samiT}. aTisa da Svidis sxvaoba 

risi tolia? {samisa}. 

aseve SevadarebT aTsa da rvas, aTsa da oTxs, aTsa da cxras. 

daskvna gameordeba: _ aTi imdeniTaa cxraze meti, 

ramdenicaa aTisa da cxris... ra? {sxvaoba} diax, sxvaoba 

gviCvenebs, Tu didi ricxvi ramdeniTaa meti, vidre patara! 

_ axla daxedeT suraTs (76.2). ra raodenobania Sedarebuli 

pirvel, yvelaze marcxena svetSi? {rva da oTxi} ras gviCvenebs 

es pirveli sveti? {rva metia, vidre oTxi, oTxiT; rva oTxze 

oTxiT metia} risi tolia rvisa da oTxis sxvaoba? {oTxisa} 

axla Tqven TviTon daawyvileT Semdegi raodenobani. ras 

gviCvenebs meore sveti? {rva metia eqvsze oriT} risi tolia 

rvisa da eqvsis sxvaoba? (da a. S.). 

------------------------------------------------------------------------------------ 

III-8. nsvef! yb{j! bov! nsvej/ wina TavSi vaswavleT texili, 

viTarc mravalkuTxedis sazRvari. axla aseve vaswavliT mrude xazs, 

viTarc saTanado nakvTis sazRvars. hb{pnwb/ sanam TviT gazomvas 

daviwyebdeT, moswavleebma unda gaiazron, Tu ra aris sigrZe da ra 

Tvisebebi aqvs mas. uamisod gazomva meqanikuri procesi iqneba. 

Tavidanaa sigrZis met-nakleboba, toloba, misi Cawera. sigrZis

- 220 - 

pirveli Tviseba (simetriuloba). 

ramdenime safexuris Semdeg saWiro iqneba Sesadarebel sigrZeTa 

aRniSvna qarTuli asoebiT da Sedegis dafaze Cawera (wers mxolod 

masw.-i da ara moswavleebi). magaliTad: m > s `magidis sigrZe 

metia saxazavis sigrZeze~. amitom masw.-ma unda gaiTvaliswinos: am 

droisaTvis klasma icis Tu ara saTanado asoebis cnoba. sakmarisia 

mxolod calkeuli asoebis cnoba. araa saWiro arc is, rom 

moswavlem icodes kiTxva (rac gacilebiT ufro Znelia, vidre 

asoebis cnoba) da arc is, rom icodes asoebis wera. 

------------------------------------------------------------------------------------ 

87/4/!xsjt!tb{Swbsj/!!TfebsfcjU!sUvmj!tb{Swsfcj/ 

_ ra aris daxazuli ruxad am suraTze? {rvakuTxedi da 

wre} SevadaroT erTmaneTs maTi sazRvrebi. ra aris rvakuTxedis 

sazRvari? {Sekruli texili} ramdeni monakveTisagan 

Sedgeba igi? {rva monakveTisagan} ramdeni wvero aqvs am texils? 

{rva} ramdeni bolo aqvs? {arcerTi, nuli} ratom? 

{imitom, rom Sekrulia} riT gansxvavdeba rvakuTxedis sazRvrisagan 

wris sazRvari? {ar Sedgeba monakveTebisagan, ar aris 

texili} axla gverdiT vardisferi nakvTebi naxeT. aris Tu ara 

vardisferi wre da ruxi wre erTmaneTis toli? {aris} 

vardisferi rvakuTxedi da ruxi rvakuTxedi? {esenic tolia} 

axla vardisferi nakvTebis sazRvrebi SeaferadeT Savad! 

_ kidev ra aris daxazuli aq? ruxi nakvTebis marjvniv? 

{Sekruli texili} ise gaaferadeT, rom es texili gaxdes 

mravalkuTxedis sazRvari! winaswar kargad daakvirdiT, ar 

SegeSaloT gaferadeba! (unda gaaferadon texilad Caklaknili 

zoli!) ramdenkuTxedi miiReT? {aTkuTxedi} axla gverdiT 

calke amoxateT mxolod sazRvari! ramdeni wvero eqneba am 

texils? {aTi} ramdeni bolo wertili eqneba? {arcerTi, 

nuli} ratom? {imitom, rom Sekrulia}. 

87/5/! `txsbgj! qbtvyfcj! `! fsUnbofUt~/ masw.-i moswavles 

etyvis Sekrebisa an gamoklebis magaliTs, oRond terminebiT 

`Sesakrebi, saklebi, maklebi.~ moswavlem momdevno moswavles 

unda dausvas Sesabamisi SekiTxva. momdevno moswavlem ki unda

- 221 - 

gamoTvalos. mis Semdeg, mesame moswavles kvlav masw.-i 

Camouyalibebs magaliTs, mesame moswavle ki meoTxes dausvams 

saTanado SekiTxvas. meoTxem unda gamoTvalos. 

nimuSebi: _ Sesakrebebia ori da rva. dausvi SekiTxva dianas! 

{risi tolia jami?} diana, gvipasuxe! {aTisa} kargia. axla 

dimitri, saklebia cxra, maklebia xuTi. (dimitri ekiTxeba 

momdevno moswavles: _ risi tolia sxvaoba?). da ase Semdeg. 

g 77. fsUj!dbmj!kvgUj/!shpmjt!tb{Swbsj/!sbnefoj!blmjb!/// 

88/2/!fsUj!dbmj!kvgUj;!1 · 3, 1 · 7, 1 · 2!!eb!tywb/ 

jer swrafad gavameorebinebT 75.1-sa da 76.1-s. Semdeg masw.-i 

saSualo moswavles daavalebs: _ gamodi da daawyve kubebi, 

erTi cali sameuli! 

Tan dafaze dawers 1 · 3. Semdeg dawers 1 · 5 da sxvas 

daavalebs amdeni raodenobis marTkuTxedebis daxazvas. Semdeg 

klasi msr-Si (77.1) TviTeul CarCos xaziT SeaerTebs 

Sesabamis CanawerTan (gamosaxulebasTan) qveda striqonSi. 

_ maS, ramdeni yofila erTi cali xuTeuli? {es yofila 

imdenive, ramdenicaa xuTi, erTi cali xuTeuli igivea, rac 

xuTeuli, xuTi}. 

_ axla vin mixvdeba, romelia meti da ramdeniT: 

4 · 1 Tu 1 · 4 (es gamosaxulebani dafazea dawerili)? 

Tuki gauWirdebaT, jer xmamaRla waikiTxon gamosaxulebani 

da Semdeg Seadaron {tolia, oTxi cali erTeuli igive raodenobaa, 

rac erTi cali oTxeuli, orives raodenoba oTxia!}. 

88/3/! sbnefoj! blmjb! sjdywt! nfpsf! sjdywbnef! tywbpcbtUbo! 

eblbwTjsfcjU. auCqareblad gavimeorebT 76.2-s: mTw-dan 

oceulis TvalsaCinoebiT (gv.17) iseve SevadarebT erTmaneTs, 

magaliTad, rvaeulsa da aTeuls, Semdeg aTeulsa da Svideuls. 

daskvnac orivejer unda Camoyalibdes. 

Semdeg mTw gadaiSleba `10~-ze. 

_ rogoraa am jamSi dawyobili aTi wre? {Svidi da sami} 

SevadaroT erTmaneTs Svidi da aTi. ramdeni aklia Svids 

aTamde? {sami} saidan ici? {aTi tolia Svidisa da samis

- 222 - 

jamisa. Svids aTamde aklia sami} ramdeniTaa aTi Svidze meti? 

{samiT} risi tolia aTisa da Svidis sxvaoba? {samisa}. 

aseve `sammagad~ Camoyalibdeba ramdenime Sedareba: aTi da 

eqvsi, aTi da xuTi, cxra da aTi. 

ebtlwob/ patara ricxvs did ricxvamde aklia imdeni, 

ramdenicaa didisa da pataris sxvaoba. 

88/4/! shpmjt! tb{Swbsj/! TfebsfcjU! sUvmj! tb{Swsfcj/ 

_ naxazze (77.3) wyvetilad daxazulia rgolis sazRvari. gaaferadeT 

ise, rom gamoCndes, romeli rgolis sazRvaria es. axla 

gverdiT rom vardisferi rgolia, misi sazRvari lurjad 

SeaferadeT! rogori sazRvari hqonia rgols _ mTliania Tu 

ara igi? {ara, ori cal-calke nawilisagan Sedgeba} saxazaviT 

daxazeT kidev erTi rgoli, jer sazRvari dauxazeT da mere 

gaaferadeT! 

_ axla daakvirdiT lokokinasaviT Caxveul nakvTs. 

gaaferadeT ise, rom gamoCndes, Tu romeli nakvTis sazRvaria. 

jer gaferadeba ar daiwyoT, jer kargad daakvirdiT! axla 

gverdiT daxateT amdagvari nakvTi. jer sazRvari dauxazeT da 

mere gaaferadeT! 

88/5/!`lsjbybt!xjxjmfcj~/ imdagvarivea, rogorc 72.4. oRond 

cota sxva rameebi xdeba _ mokle-mokle CanarTebs Tavad masw.-i 

moifiqrebs. 

arsebiTi ki igivea: aTis daSla SeskrebTa jamebad, ramdenimes 

mimateba Tu mokleba, mimarTeba `SigniT\gareT\sazRvarze~. 

g 78. kbnjt!Tfebsfcb!sjdywUbo/!!nsvef!yb{j!bov!nsvej/!! 

89/2/!fsUjebjnbwf!hspwjt!ebUwmb!tywbebtywb!kvgUfcbe!! 

)vobTUpe*/! magnitur dafazea 8 rgolis grova. gamova erTi 

moswavle da daawyobs wyvilebad. Sedegs ityvis da masw.-i 

dawers: 4 · 2. gamova meore moswavle, dawyobils aurevs 

da daawyobs oTxeulebad. Sedegs ityvis da masw.-i dawers 

pirveli Canaweris qvemoT: 2 · 4. 

_ axla adgilidan miTxras zazam: es grova rom 

erTeulebad davTvaloT, ras miviRebT? {rva cal erTeuls}

- 223 - 

davweroT! (masw.-i ori Canaweris qveS dawers: 8 · 1). adgilidan 

miTxras cialam: es grova rom rvaeulebad davTvaloT, ras 

miviRebT? {erT cal rvaeuls} davweroT! (masw.-i sami Canaweris 

qveS dawers: 1 · 8). maS, es erTi grova ramdennairad 

davawyveT? {oTxnairad} waikiTxeT oTxive! (waikiTxaven} diax, 

raodenoba rvaa, esaa rva cali erTeuli. igivea erTi cali 

rvaeuli. magram SeiZleba davTvaloT wyvilebad da miviRebT 

oTx cal wyvils. SeiZleba davTvaloT oTxeulebad da 

miviRebT... ras? {or cal oTxeuls}. es raodenobaa meti Tu 

es? (masw.-i rig-rigobiT miuTiTebs Canawerebze). es Tu es? 

{tolia, esenic tolia, esenic} saidan ici? {xom erTsadaimave 

grovas vawyobdiT} grovis sxvanaiarad gadalagebiT misi 

raodenoba SeiZleba Tu ara rom Seicvalos? {ara, ar SeiZleba} 

diax, raodenoba igive rCeba! 

_ axla daakvirdiT suraTs (78.1). risi grovaa marcxniv? 

{yvavilis naxatebisa). SemoxazeT lurji xazebiT ise, rom 

sameulebi miiRoT! {unda Semoixazos svetebi} ramdeni sameuli 

miviReT? {eqvsi} davwer: 6 · 3. axla SemoxazeT mwvane xazebiT 

ise, rom eqvseulebi miiRoT! {unda Semoixazos striqonebi} 

ramdeni eqvseuli miviReT? {sami} davwer: 3 · 6. axla 

SemoxazeT wiTeli xazebiT ise, rom wyvilebi miiRoT! ramdeni 

wyvili miviReT? {cxra} davwer: 9 · 2. gveyofa, Torem mTlad 

aWrelda da veRarafers davinaxavT. maS, yvavilebis naxatTa es 

erTi grova ramdennairad davTvaleT? {samnairad} ai me aq 

Cavwere samive. waikiTxeT! {waikiTxaven}. 

_ marjvniv ki lokokinasmagvari mrude xazebis grovaa. 

rogor SeiZleba davTvaloT es grova? {sameulebad} SemoxazeT 

mwvane xazebiT! ramdeni sameulia? {sami sameulia} davwer: 3 · 

3. kidev rogor SeiZleba davTvaloT? {erTeulebad} ramdeni 

erTeulia? {cxra} davwer: 9 · 1. kidev rogor SeiZleba davTvaloT? 

{cxraeulebad} ramdeni cxraeulia? {erTi} davwer: 1 · 9. 

maS, es erTi grova ramdennairad davTvaleT? {samnairad} ai me 

aq Cavwere samive. waikiTxeT! {waikiTxaven}. 

89/3/! kbnjt! Tfebsfcb! sjdywUbo/! ! _ daakvirdiT marcxena

- 224 - 

svets (78.2). romelia is pirveli Sesakrebi, romelic sul 

meordeba? {oTxi} CamoTvaleT, meore Sesakrebebi romeli 

ricxvebia {erTi, ori, sami, oTxi, xuTi} axla pirveli, sul 

zeda jami SevadaroT Svids. romelia meti? {Svidi} ratom? 

{oTxs mimatebuli erTi tolia xuTisa da xuTi naklebia, vidre 

Svidi} CawereT saWiro niSani! (Caweren < niSans). meore jami 

SevadaroT Svids. romelia meti? {Svidi} ratom? {4+2=6 da 

eqvsi naklebia, vidre Svidi} CawereT saWiro niSani (Caweren < 

niSans). mesame jami SevadaroT Svids. romelia meti? {tolia} 

ratom? {4+3=7} CawereT saWiro niSani! (Caweren = niSans). 

meoTxe jami SevadaroT Svids. romelia meti? {jami} ratom? 

{4+4=8, 8>7} CawereT saWiro niSani! (Caweren > niSans). mexuTe 

jami SevadaroT Svids. romelia meti? {jami} ratom? {4+5=9, 

9>7} CawereT saWiro niSani! (Caweren > niSans). 

danarCen or svetSi saWiro naSnebs Caweren Sin, saSinao 

davalebad. 

89/4/! nsvef! yb{j! bov! nsvej/! ! hbytojmj! nsvejt! cpmpfcj! 

)cpmp! xfsujmfcj*/! dafaze daxazulia raime uswormasworo 

nakvTi, romlis sazRvari nawilobriv texilia da nawilobriv 

_ mrudi. _ gamovides irakli da am nakvTis sazRvari calke 

gadmoxatos aqeT! diax, esaa. amgvar xazs ewodeba mrude xazi, 

anu mrudi. mrudi. riT gansxvavdeba mrudi da texili? 

{texili mxolod monakveTebisagan Sedgeba, mrudi ki _ ara} 

Sekruli texili risi sazRvaria? {mravalkuTxedisa} magali- 

Tad, aTi monakveTisagan Sedenili texili romeli mravalkuTxedis 

sazRvri iqneba? {aTkuxedisa} mrudi Tu aris mravalkuxedis 

sazRvari? {ara} SeiZleba Tu ara, rom zogjer 

mravalkuTxeds sazRvari mrudi hqondes? {ara, maSin igi ar 

iqneboda mravalkuTxedi!} es mrudi rogoria _ Sekruli Tu 

gaxsnili? {Sekruli}. 

_ axla daaTvaliereT suraTi (78.3). ra aris daxazuli? 

{mrudi xazebi} ramdeni? {oTxi} rogoria marcxena, pirveli 

mrudi? {Sekrulia} gaaferadeT ise, rom igi romeliRac nakv- 

Tis sazRvari gaxdes! meore mrudi rogoria? {gaxsnilia} igi

- 225 - 

ver iqneba nakvTis sazRvari. mesame mrudi? {isic gaxsnilia da 

ver iqneba nakvTis sazRvari} meoTxe? {Sekrulia} gaaferadeT 

ise, rom igi romeliRac nakvTis sazRvari gaxdes! ras hgavs 

miRebuli nakvTi? {wylis wveTs} kargi. romelime am oTxi 

mrudidan Tu Seicavs monakveTs? {ki, mesame mrudi Seicavs}. 

axla cariel adgilas daxazeT axali, sul sxvagvari mrudi! 

89/5/ Sejibri simaRlis mixedviT dadgomaSi _ ix. 5.4. 

g 79. Ubwj!21/2.jt!hbnfpsfcb.!sjdywjt!Tfebsfcb!kbnUbo/! 

8:/2/!Ubwj!21/2.jt!Tfkbnfcb!eb!hbnfpsfcb. 

keTdeba orgvari savarjiSoebi: 1) masw.-i dafaze dawers 

gamosaxulebas (magaliTad: 7 · 3, 7 · 1, 2 · 9, 1 · 6 da kidev 

ramdenime) da moswavleebi dafaze an magadaze awyoben 

Sesabamisad nivTebs (an dafaze xataven). orma-samma moswavlem 

erTad imuSaos. Sedegs sxva moswavle waikiTxavs adgilidan. 

2) dafazea erTidaigive 30-iani grova da mas daawyoben xan 

eqvseulebad, xan sameulebad, xan xuTeulebad, xan aTeulebad. 

Sedegebi dafaze daiwereba da moswavleebi waikiTxaven. 

bolos, suraTze (79.1) borjRalebis grovas jer 

Semoxazvis gareSe daTvlian: xuTeulebad {oTxi xuTeulia}, 

Semdeg _ aTeulebad {ori aTeulia}, Semdeg _ wyvilebad 

{svetebs daTvlian _ aTi wyvilia}, Semdeg ki wiTeli xazebiT 

Semoxazavan ise, rom miiRon oTxeulebi _ am dros aRar unda 

miaqcion yuradReba lurj gamyof xazs! miiReben xuT oTxeuls. 

8:/3/! sjdywjt! Tfebsfcb! kbnUbo/ am safexurs cota dro 

eyofa. yvelaferi keTdeba iseve, rogorc 78.2-Si. 

_ daakvirdiT marcxena svets (79.2). romelia is meore 

Sesakrebi, romelic sul meordeba? {ori} CamoTvaleT, pirveli 

Sesakrebebi romeli ricxvebia {ori, sami, oTxi, xuTi} axla 

eqvss SevadaroT pirveli, sul zeda jami... 

da ase Semdeg _ iseTive SekiTxvebi daismis da imave 

TanmimdevrobiT, rogorc 78.2-Si. 

danarCen or svetSi saWiro naSnebs Caweren Sin, saSinao 

davalebad.

- 226 - 

8:/4/!`vgsp!hs[fmj~!eb!`vgsp!nplmf~!`!ebupmfcjU/ 

am safexurs cota dro eyofa. klass jer gavameorebinebT 

sivrciT mimarTebas maRali \ dabali: jer moswavleebis 

simaRleSi datolebiT, Semdeg ki warmosaxviT, magaliTad: ra 

ufro maRalia, Zroxa Tu didi adamiani? Cveni skolis ezoSi 

mdgari buCqi Tu es karada? 

Semdeg moswavleebs rig-rigobiT SevadarebinebT sxvadasxva 

nivTebs ukve sigrZiT. Sesadarebeli wyvilebi ise SeirCeva, rom 

sigrZeTa gansxvaveba ar iyos Zalian mkveTri, raTa saWiro 

gaxdes maTi datoleba. _ es joxi ufro grZelia Tu 

saxazavi? ar Cans, xom? ra vqnaT? {davatoloT} es saxazavi 

ufro moklea Tu magida? {davatoloT} es daxazuli monakveTi 

ufro grZelia Tu saxazavi? (da ase Semdeg _ xan iTqmis 

`grZeli~, xan _ `mokle~). 

8:/5/! `DyjsfcjU! bhfcboj~/ moswavlebs yvela merxze 

dawyobili aqvT Cxirebi an asanTis Rerebi. merxze msxdomi ori 

moswavle erTad asrulebs masw.-is davalebebs: _ avagoT gemi 

eqvsi CxiriT! saxli aTi CxiriT! (da sxva _ masw.-s winaswar 

unda hqondes mofiqrebuli sqemebi). 

8:/:/ msr-s gadavaSlevinebT 56.6-ze, SevaxsenebT, rom iq 

iyo `nakvTebis Sekreba~. aq ki `nakvTebis gamoklebaa~. kanonzomierebas 

moswavleebi Tavad unda mixvdnen. vinc ver mixvdeba, 

momdevno gakveTilze klasSi gaigebs da Seavsebs kidec. 

g 80. obTU/!kvgU/!ebUwmb/!ùpmj!tjhs[f~!`!ebupmfcjU/ 

91/1/ 79.7-Si saWiroa damatebiTi SekiTxva: marcxena 

CarCoSia ufro meti jvrebi Tu marjvenaSi? {tolia, radganac 

suraTebi zustad erTnairia} ese igi, erTidaigive grova jer 

davTvaleT wyvilebad, Semdeg _ oTxeulebad. 

------------------------------------------------------------------------------------ 

I.11. obTUjboj! kvgUfcj! eb! dofcb! òbTUj~/ xdeba naSTiani 

SemTxvevebis dawyoba, Cawera da wakiTxva. kerZo SemTxvevaa is, roca 

naSTia 0, anu roca unaSTod daewyoba (rac ukve viswavleT wina 

gakveTilebze).

- 227 - 

bolos, gaiazreba Zneli sakiTxi: naSTi ar SeiZleba iyos 

jufTze meti. jer-jerobiT (Tav 12-mde), arsad ar daismis sakiTxic 

ki imis Sesaxeb, Tu sul ramdeni nivTia jufTebSi (erTad). 

naSTis cnebis Sesaxeb ix. zms $ 3. iseve rogorc adre, zogad 

termins `jufTi~ klasSi arcerTxel ar vaxsenebT. igi mxolod 

masw.-isTvisaa. klasSi vityviT mxolod konkretulad: oTxeuli, 

aTeuli, cxraeuli, wyvili da sxva romelic dagvWirdeba. 

------------------------------------------------------------------------------------- 

91/2/!xjobtxbs!ebxzpcjmj!obTUjboj!kvgUfcjt!ebUwmb;!`tbnj! 

dbmj!pUyfvmj!eb!ljefw!psj~/!!!termini `naSTi~ am gakveTilze 

ar unda vaxsenoT! 

masw.-i dafaze dawers: 6 · 3. _ gamovides amirani da daawyos 

amdeni wre! (gamova da daawyobs). axla me davumateb kidev or 

wres (daumatebs gverdiT). axla vin daTvlis sameulebad 

dafaze wreebs? {eqvsi cali sameuli da kidev ori}. diax, da 

kidev oria! 

masw.-i amas kidev erTxel gaimeorebs, nel-nela, Tan 

TviTeuli sityvis warmoTqmisas miuTiTebs saTanado jufTebs. 

Semdeg dafaze daxatavs 8 · 2 + 1 raodenobis monakveTebs 

da moswavleebi adgilidan daTvlian wyvilebad. 

_ axla vnaxoT suraTze (80.1). ramdeni CarCoa? {sami} rogoraa 

pirvel CarCoSi dawyobili Savi wreebi? {oTxeulebad} 

daTvaleT oTxeulebad! {sami cali oTxeuli da kidev ori}. 

aseve daTvlian meore CarCoSi {xuTi cali sameuli da kidev 

erTi} da mesame CarCoSi {sami cali xuTeuli da kidev 

sami}. 

91/3/! tywbpcjt! Tfebsfcb! sjdywUbo/! am safexurs cota dro 

eyofa. yvelaferi keTdeba iseve, rogorc 78.2-Si. 

_ daakvirdiT marcxena svets (80.2). romelia is saklebi, 

romelic sul meordeba? {cxra} CamoTvaleT, maklebebi romeli 

ricxvebia {nuli, erTi, ori, sami, oTxi} axla pirveli, sul 

zeda sxvaoba SevadaroT Svids... 

da ase Semdeg _ iseTive SekiTxvebi daismis da imave 

TanmimdevrobiT, rogorc 78.2-Si, oRond, cxadia, Sekreba

- 228 - 

icvleba gamoklebiT. 

danarCen or svetSi saWiro niSnebs Caweren Sin, saSinao 

davalebad. 

91/4/! ùpmj! tjhs[f~! `! ebupmfcjU/! Sedardeba sxvadasxva 

sigrZeebi, iseve, rogorc 79.3-Si, oRond erTi siaxliT: zogjer 

Sesadarebeli sigrZeebi toli aRmoCndeba. masw.-s es winaswar 

unda hqondes momzadebuli (saTanado sigrZis joxi, merxis 

toli sigrZis Toki, dafaze daxazuli monakveTi da sxva). 

daismis SekiTxva: _ Cvens saklaso oTaxSi visi merxia yvelaze 

grZeli? {aravisi, yvela merxis sigrZe erTnairia, tolia}. 

bolos naxazze (80.3) TvalzomiT movaZebninebT or tol 

monakveTs {pirveli da meoTxea}. 

91/5/! `yfmfcjt! tjhs[jt! njyfewjU! ebehpnb~/ gamova ori 

moswavle, sworad gaSlian xelebs da daetolebian _ vis ufro 

grZeli aqvs. visac ufro grZeli aqvs, dadgeba ufro marcxniv. 

gamova Semdegi moswavle da daetoleba _ Tavis adgils 

moZebnis. gamova meoTxe da _ isic. ase dalagdeba mTeli klasi 

xelebis sigrZis mixedviT. 

Tuki zog or moswavles toli sigrZis xelebi aRmoaCndeba, 

erTmaneTis win dgebian, ese igi, rigSi erTsadaimave adgilas. 

Zalian bevri datoleba rom ar gaxdes saWiro, axalgamosul 

moswavles masw.-i miuTiTebs, Tu romel moswavleebs 

daetolos _ daetolos daaxloebiT misi simaRlis 

moswavleebs, ufro maRlebsa da ufro dablebs nu daetoleba. 

bolos, yuradRebas mivaqcevT imas, xelebis sigrZis mixedviT 

dadgoma igivea Tu ara, rac simaRlis mixedviT dadgoma. 

91/8/ unda Seasrulon TvalzomiT! 

g 81. òbTUj~/!tjhs[jt!nfu.oblmfcpcjt!Dbxfsb!ebgb{f/! 

92/2/!ufsnjoj!òbTUj~/!`tbnj!dbmj!pUyfvmj!eb!obTUj!psj~/ 

yvelaferi keTdeba iseve, rogorc 80.1-Si, oRond masw.-s 

Semoaqvs termini `naSTi~. _ diax, aqaa sami xuTeuli da kidev 

oTxi. ai am oTxs hqvia naSTi. xuTeuli samia, naSTad ki ra 

rCeba? {oTxi}.

- 229 - 

_ axla vnaxoT suraTze (81.1). ramdeni CarCoa? {sami} 

rogoraa pirvel CarCoSi dawyobili Savi wreebi? {eqvseulebad} 

risi tolia naSTi? {xuTisa} daTvaleT eqvseulebad! 

{sami cali eqvseuli da naSTi xuTi}. 

aseve daTvlian meore CarCoSi {erTi cali cxraeuli da 

naSTi Svidi} da mesameSic {oTxi cali wyvili da naSTi erTi}. 

92/3/! sjdywjt! Tfebsfcb! tywbpcbtUbo/! ! ! yvelaferi keTdeba 

iseve, rogorc 80.2-Si, oRond ricxvi Sedardeba sxvaobas. 

Tanac, pirvel da meore svetebSi ara saklebebia erTidaigive, 

aramed _ maklebebi. 

92/4/!tjhs[jt!nfu.oblmfcpcjt!Dbxfsb!ebgb{f/!ojnvTj;!b >!j/ 

yvelaferi keTdeba iseve, rogorc 80.3-Si, oRond erTi siaxliT: 

Sesadarebeli sagnebis sigrZes aRvniSnavT raime asoTi da Sedarebis 

Sedegs dafaze dawers masw.-i. magaliTad: _ am magidis 

sigrZe aRvniSnoT m asoTi, Tokis sigrZe ki T asoTi. romeli 

iyo ufro grZeli? {toli iyo} diax, davweroT: m = T. ras 

gveubneba es Canaweri? {magidis sigrZe tolia Tokis sigrZisa}. 

aseve Caiwereba: s < k {saxazavis sigrZe naklebia karis 

sigrZeze} da sxva. 

92/5/!`nbUfnbujlvsj!tjuzwfcjt!ebtbyfmfcb~/ ix. 38.4 

(am droisaTvis damatebulia mravali axali sityvac!). 

g 82. obTUjboj!kvgU/!Dbxfsb/!sblj!b >!j-!!bnjupn!!j

- 230 - 

emateba! amitom Caiwereba mimatebis, Sekrebis niSniT! 

bolos, moswavleebi msr-Si (82.1) TviTeul CarCos xaziT 

SeaerTeben Sesabamis CanawerTan (gamosaxulebasTan). 

93/3/! kbnjtb! eb! sjdywjt! Tfebsfcb! tbzpgbdypwsfcp! bnpdb. 

ofcTj/! jer Sesruldeba msr-s 82.2. qveda striqonSi weria 

jamebi. zemoT ki jamis naxatia. _ ratom weria marcxniv 3+6? 

{imitom, rom mis zemoTaa sami marTkuTxedi da eqvsi samkuTxedi, 

erTad Sekrebili, SeerTebuli} axla Tqven TviTon CaxateT 

saWiro raodenobis nakvTebi meore, cariel jamSi. radara 

nakvTebi SevarCioT? kargi, iyos monakveTebi da wreebi. CaxateT, 

oRond kargad daTvaleT, ar SegeSaloT! 

aseve Seivseba mesame `jamic~. 

Semdeg amoixsneba ramdenime sayofacxovrebo amocana. 

_ ori gundi fexburTs TamaSobda. pirveli gundidan giorgim 

gaitana 5 burTi, meti aravis ar gautania. meore gundidan 

nikam gaitana 2 burTi, xolo levanma _ 4. romel gunds 

gaumarjvia? ratom? 

rogorc Cveulebriv, jer gavameorebinebT pirobas. 

_ ramdeni burTi gaitana pirvelma gundma? {xuTi} (masw.-i 

dafaze dawers 5). ramdeni burTi gaitana meore gundma? 

{ori da sami} (masw.-i dafaze dawers 2 + 4). romels gautania 

meti burTi? {meore gunds} ratom? (2+4 = 6 da 6 > 5). modi, 

giorgi, Cawere saWiro niSani! 

giorgi dafaze 5-iansa da jams Soris Cawers < niSans, 

jamis mniSvnelobas 6 ki ar dawers. amas arc masw.-i dawers. 

saboloo Canaweria: 5 < 2 + 4. 

_ romel gunds gaumarjvia? ratom? ramdeniT? {erTiT}. 

aseve amoixsneba meore amocana: 

_ lia, ia, Tea da gia sokos agrovebdnen. liam ipova 2 

soko (dawers dafaze _ 2), iam _ amdenive. ramdeni upovia ias? 

{ori} (masw.-i pirveli cifrisgan cota moSorebiT dawers _ 2). 

Team ipova 3 soko (dawers cota moSorebiT _ 3), giam _ 

Svidi (dawers cota moSorebiT _ 7). gogonebs ufro meti 

soko upoviaT Tu gias?

- 231 - 

amoixsneba wina amocanis magvarad. dafaze saboloo Canaweri 

iqneba: 2 + 2 + 3 = 7. 

_ ras da ras vadarebdiT am amocanebSi? raRac ricxvs 

vadarebdiT... ras? {jams} diax, ricxvs vadarebdiT jams. 

xvalisTvis Tqven TviTon moigoneT axali amocana, romlis 

amosaxsnelad saWiro iqneba jamisa da ricxvis Sedareba. 

oRond, Tqveni amocanebi sxva raRacebis Sesaxeb unda iyos! 

93/4/! Uvlj! yb{j! `b~! vgsp! hs[fmjb-! wjesf! `j~-! nbTjo! `j~! 

vgsp!nplmf!zpgjmb-!wjesf!`b~/!!sblj!b >!j-!!bnjupn!!j k. amitom ufro mokle romeli 

yofila? {kari} diax, davweroT: k < T. 

bolos, masw.-i klass warmodgeniT Seadarebinebs sigrZeebs. 

_ texili xazi ufro grZeli iyo, vidre mrude xazi. 

davwer: t > m. ese igi, mrude xazi ufro... ra? {ufro mokle 

yofila, vidre texili} diax, davweroT: m < t. 

aseve _ Sekruli da gaxsnili xazebis Sesaxeb: 

raki S > g, amitom g < S. 

93/5/! èbwehfU! Dbobxfsjt! njyfewjU~/ ix. 74.4, oRond erTi 

siaxlea: masw.-i wers naSTian gamosaxulebebs: 

5 · 4 + 1, 1 · 9 + 7 da sxva. 

93/9/ unda gakeTdes 82.2-is magvarad. am amocanis garCevisas 

ki moswavleebma sityvieradac unda Camoayalibon: pirvel 

`jamSia~ 3 wertili da 5 monakveTi; meoreSi _ 4 gaxsnili 

texili da 3 Sekruli texili; mesameSi _ 5 wre da 4 

mravalkuTxedi. 

g 83.!obTUjboj!kvgU/!ebxzpcb/!Uvlj!p !p/ 

94/2/! obTUjboj! kvgUfcjt! ebxzpcb! xjobtxbs! npdfnvmj! 

Dbobxfsjt!njyfewjU/!masw.-i dafaze dawers: 4 · 4 + 2. gamovides 

moswavle da daawyos amdeni rgoli. sxva moswavlem ki 

waikiTxos.

- 232 - 

amdagvari ram gakeTdeba kidev orjer, oRond nivTebis 

dawyoba Seicvleba nakvTebis dafaze daxatviT. 

Semdeg Sesruldeba msr-s 83.1. _ ramdeni CarCoa aq 

daxazuli? (xuTi). pirvel, sul zeda CarCoSi dasviT imdeni 

wertili, ramdenic am CarCos uweria marcxniv! aseve _ meore 

CarCoSic. danarCenebs Seavseben Sin, saSinao davalebad. 

94/3/! tywbpcjtb! eb! sjdywjt! Tfebsfcb! tbzpgbdypwsfcp! bnp. 

dbofcTj. _ vaJas 9 fanqari hqonda, fiqrias _ 7. Semdeg vaJas 

oTxi fanqari gauTavda. axla romels aqvs meti fanqari? 

ramdeniT? 

amoixsneba iseve, rogorc 82.2-is amocanebi. dafaze 

saboloo Canaweri iqneba: 9 – 4 < 7. _ ramdeniT? {fiqrias 

aqvs oriT meti fanqari}. 

2) ezoSi 5 bavSvi iyo, saburTao moedanze _ 9. Semdeg 

moednidan 3 bavSvi Sin wavida. axla sad ufro naklebi bavSvia? 

ramdeniT? { 5 < 9 – 3, naklebia 1-iT}. 

3) dafaze daxazuli iyo 8 texili da 5 mrudi. Semdeg 3 

texili waSales. axla romelia ufro meti? ramdeniT? 

{8 – 3 = 5. imdenivea, raodenoba tolia}. 

_ sul ramdeni amocana amovxseniT dRes? {sami} gaixseneT, 

ris Sesaxeb iyo pirveli amocana {bavSvebis fanqrebis Sesaxeb}. 

meore? {bavSvebi ezoSi da moedanze} mesame? {dafaze daxazuli 

texilebisa da mrudebis Sesaxeb}. ras da ras vadarebdiT samive 

am amocanaSi? raRac ricxvs vadarebdiT... ras? {sxvaobas} diax, 

ricxvs vadarebdiT sxvaobas. xvalisTvis Tqven TviTon 

moigoneT axali amocana, romlis amosaxsnelad saWiro iqneba 

sxvaobisa da ricxvis Sedareba. oRond, Tqveni amocanebi sxva 

raRacebis Sesaxeb unda iyos! 

94/4/! Uvlj! p !p/!(cxadia, klasSi gaSlili 

msjelobiT). yvelaferi keTdeba iseve, rogorc 82.3-Si, oRond 

erTi araarsebiTi cvlilebiT: icvleba met-naklebobis 

mimarTuleba _ jer sigrZis naklebobaa (es ufro moklea), 

Semdeg _ metoba (amitom is ufro grZelia). 

94/5/!Ùwbmjt!ebybnybnfcb~. ix. 15.4.

- 233 - 

94/6 iseve rogorc, magaliTad, 79.5-Si, saWiroa mxolod 

gadawera. cxadia, araviTari angariSi. 

g 84. hspwjt!ebUwmb!obTUjbo!kvgUfcbe/!U/!:/3.jt!hbnfpsfcb/ 

95/1/ 83.7-is garCevisas moswavleebma sityvieradac unda 

Camoayalibon: pirvel `jamSia~ 2 oTxkuTxedi da 5 aTkuTxedi; 

meoreSi _ 6 oTxkuTxedi da 3 samkuTxedi; mesameSi _ 4 

gaxsnili mrudi da 3 Sekruli texili. 

95/2/!hspwjt!ebUwmb!obTUjbo!kvgUfcbe!`!Tfefhjt!DbxfsjU/ 

dafaze daxazulia mrudebis 23-iani grova. _ beso, daTvale 

xuTeulebad! ra miiRe? axla CavweroT Sedegi. rogor 

davwero? {4 · 5 + 3}. qeTevan, waikiTxe! 

Sesruldeba kidev ramdenime amgvari savarjiSo, masw.-is mier 

Sedegis CaweriT. 

Semdeg msr-Si (84.1) gaakeTeben dalagebul raodenobaTa 

Sesabamis Canawerebs. Seavseben mxolod pirvel striqons 

(cxadia, masw.-i werasac miaqcevs yuradRebas), meore _ saSinao 

davalebad eqnebaT. 

95/3/! Ubwj! :/3.jt! hbnfpsfcb! eb! Tfkbnfcb/! SedarebiT swrafad 

amoixsneba xuTi amocana. isini 82.2-83.2-is magvarad 

damuSavdeba. 

1) dReobaze gogonebs namcxvrebi dauriges. rusudans misces 

9 namcxvari, nestans _ 4. nestanma yvela nacxvari SeWama, 

rusudanma ki 4 namcxvari ZmisTvis Seinaxa, danarCeni SeWama. 

romel gogonas ufro meti namcxvari SeuWamia? ramdeniT? {9 – 

4 > 4, metia 1-iT}. 

2) dafaze marcxniv daxazulia 3 kvadrati da 8 wre. 

marjvniv ki daxazulia 6 samkuTxedi. romelia meti _ 

mravalkuTxedebi Tu wreebi? ramdeniT? {3 + 6 > 8, metia 

1-iT}. _ ratom Sekribe sami da eqvsi? (cxadia, yuradReba unda 

gamaxvildes imaze, rom kvadratica da samkuTxedic _ 

mravalkuTxedebia, es yvelam kargad unda gaiazros). 

3) vardis buCqze 4 kokori gaifurCqna, gaufurCqneli 

kokrebi ki 4-iT meti darCa. ramdeni gaufurCqneli kokori

- 234 - 

darCa vardis buCqze? {4 + 4 = 8} ratom unda mivumatoT 

oTxs oTxi? 

4) klasSi 9 vaJia, gogonebis raodenoba 2-iT naklebia. 

ramdeni gogonaa klasSi? 

5) sul ramdeni amocana amovxseniT dRes? {oTxi} wina 

gakveTilze sami amocana amovxseniT. ramdeniT meti amogvixsnia 

dRes? {erTiT} kidev sami amocana rom amogvexsna, ramdeni 

gveqneboda dRes amoxsnili? {4 + 3 = 7} ramdeniTaa amaze 

naklebi wina gakveTilze amoxsnili amocanaTa raodenoba? 

{oTxiT} rogor gaige? {dRes _ 7, guSin _ 3, Svids 

gamoklebuli sami udris oTxs}. 

95/4/!zwfmb!Tftb[mfcmpcjt!bnpxvswjU; sbnefoj!nsbwbmlvU. 

yfej! Tfj[mfcb! ebwjobypU! obyb{{f! )bsbvnfuft! frwtjtb*/! nbUj! 

dbm.dbmlf! bnpybuwb/! _ daakvirdiT naxazs (84.3). radara 

mravalkuTxedebis danaxva SegiZliaT am naxazze? {erTi didi 

marTkuTxedisa da sami samkuTxedisa} gverdiTve amoxateT isini 

cal-calke! magram eseni pirdapir Cans. axla moZebneT 

Camalulebi! 

mTliani naxazi dafaze daixazeba. Zlieri moswavle gamova 

da Semoxazavs Camalulebsac: ori oTxkuTxedia (trapeciebi). 

isini miiReba wiTeli samkuTxedisa da erTi vardisferi 

samkuTxedis gaerTianebiT. moswavleebi maTac cal-calke 

amoxataven. _ maS, sul ramdeni mravalkuTxedi SeiZleba 

davinaxoT am naxazze? {eqvsi} romlebi? {sami samkuTxedi, erTi 

marTkuTxedi da oric sxva oTxkuTxedi}. diax, 3 + 1 + 2 = 6 

(dafaze daiwereba). 

95/5/! `txsbgj! qbtvyfcj! `! sbpefopcbUb! Tfwtfcb~/ ix. 39.4, 

oRond pirvel wreze raodenobas avseben cxramde, meore wreze 

da momdevnoebze _ aTamde. 

g 85.!obTUj!ovmj/!!vgsptj \!vndsptj/!!besf \!hwjbo/! 

96/1/ 84.8 iseTivea, rogoricaa 84.3, oRond didi 

oTxkuTxedi didi samkuTxediTaa Secvlili. pirdapir Cans: 

erTi didi samkuTxedi, ori patara samkuTxedi da wiTeli

- 235 - 

marTkuTxedi. maT garda, SeiZleba davinaxoT Camalulebic: ori 

oTxkuTxedia (trapeciebi). isini miiReba wiTeli marTkuTxedisa 

da TiTo vardisferi samkuTxedis gaerTianebiT. 

_ maS, sul ramdeni mravalkuTxedi SeiZleba davinaxoT am 

naxazze? {eqvsi} romlebi? {sami samkuTxedi, erTi marTkuTxedi 

da oric sxva oTxkuTxedi}. diax, 3 + 1 + 2 = 6 (dafaze 

daiwereba). 

96/2/!obTUj!ovmj;!5 · 2 + 0-!bov!`yvUj!dbmj!xzwjmj!eb!obTUj! 

ovmj~/!gakeTdeba imdagvarive savarjiSoebi, rac 84.1-Si, oRond 

SigadaSig gamoereva unaSTo SemTxvevebic. amgvari magaliTis 

dros masw.-i ikiTxavs: _ ramdeni sameuli miviReT? {xuTi} 

naSTi ra dagvrCa? {araferi, naSTi ar dagvrCenia} diax, 

unaSTod daewyo sameulebad. rogor unda CavweroT? {5 · 3} 

kargi, magram naSTis Cawerac rom mindodes? ra vqnaT? ese igi, 

naSTi araferia, risi toli yofila naSTi? {nulisa} diax, 

naSTia nuli! amitom rogor CavweroT? {5 · 3 + 0} ras 

gviCvenebs es 5-iani? es wertili? es 3-iani? es +? es 0-iani? 

{rom naSTia nuli, rom naSTi ar morCenila}. 

Semdeg msr-Si (85.1) TviTeuli dalagebuli raodenobis 

qveS daiwereba Sesabamisi Canaweri (gamosaxuleba). sadac naSTi 

araa, daiwereba + 0. 

klasSi Seivseba zeda striqoni. qveda _ saSinao davalebad. 

------------------------------------------------------------------------------------ 

II-10. vnbsujwftj! espjUj! njnbsUfcboj! eb! nbUj! eblbwTjsfcb! 

sjhpcjU! sjdywfcUbo/! kbnfcjtb! eb! tywbpcfcjt! Tfebsfcb-! spdb! 

fsUfsUj!lpnqpofouj (am terminis gareSe!) fsUjebjhjwfb/ 

es sakiTxi, cxadia, ramdenime wina safexurs saWiroebs: jer 

xdeba mxolod jamebis Sedareba, roca pirveli Sesakrebia 

erTidaigive, Semdeg roca meore Sesakrebia erTidaigive da ase 

Semdeg. Sedarebis es sakiTxebi am TavSi mTavrdeba. Tumca, jamebisa 

da sxvaobebis Sedareba saswavlo wlis bolomde grZeldeba, oRond 

ZiriTadad mxolod saSinao davalebebSi. 

TandaTan moswavleebi gaiazreben, rom jamebisa da sxvaobebis 

Sedarebisas, roca erTerTi komponentia erTidaigive, araa saWiro

- 236 - 

angariSi isedac Cans, romelia meti. magaliTad, 13 – 4 > 13 – 7. 

Tumca, amis dabejiTebiT swavleba araa saWiro. umjobesia, rom 

moswavlem TviTon iswavlos es mravali savarjiSos Sesrulebis 

Sedegad. masw.-i ubralod yuradRebas gaamaxvilebinebs imaze, rom 

angariSi saWiro ar iyo da isedac SeiZleboda mixvedra, Tu 

romelia meti. 

sazogadod, esaa ara imdenad ariTmetikis, ramdenadac algebris 

Tema. am Temis mizani isaa, rom moswavlem angariSis gareSe SeZlos 

Sedareba. Torem angariSis Sedegad Sedareba aqamdec SexvedriaT. 

droiT mimarTebaTa Sesaxeb ix. zms $ 9. 

---------------------------------------------------------------------------------- 

96/3/! UwbmtbDjop! npdfnvmpcjU;! vgsptj \! vndsptj/! besf\! 

hwjbo/! _ Tu iciT, vin aris axla klasSi myof adamianTagan 

yvelaze ufrosi? {masw.-i} yvelaze umcrosi vin aris? {uCa} 

moswavleebs Soris vin aris yvelaze ufrosi? {elene} elisoa 

umcrosi Tu elene? {eliso}. 

_ eliso ufro adrea dabadebuli Tu uCa? {eliso} yvelaze 

adre vin aris dabadebuli? {masw.-i} yvelaze gvian? {uCa} dRes 

skolaSi yvelaze gvian vin movida? Sens ojaxSi diliT 

yvelaze adre vin adga? (da ase Semdeg _ yvela SekiTxva 

TvalsaCino monacemebs emyareba). 

96/4/! zwfmb! Tftb[mfcmpcjt! bnpxvswjU; sbebsb! oblwUfcj! 

Tfj[mfcb! ebwjobypU! obyb{{f! )bsbvnfuft! frwtjtb*/! nbUj! dbm. 

dbmlf! bnpybuwb/! damuSavdeba naxazi 85.3 iseve, rogorc 84.3, 

oRond axla gvainteresebs ara mxolod mravalkuTxedebi, 

aramed yovelgvari nakvTebi. kerZod, SeiZleba davinaxoT: didi 

wre, wris sami sxvadasxva `CamonaWeri~ (ori segmenti da maT 

Soris mdebare nawili), da kidev ori `Camaluli~ nakvTi, 

romlebic miiReba mosazRvre `CamonaWerTa~ gaerTianebiT. 

amdagvaradvea 85.8-Sic. 



Semdeg _ aTis farglebSi. 

96/9. mTliani wre, naxevarwre, ruxi naxevarmTvare, wiTeli

- 237 - 

namgalamTvare, da Camaluli: ruxi naxevarmTvare da wiTeli 

naxevarwre erTad. 

g 86. nplmfe!xbljUywb-!besfhwjbo-!kfsTfnefh/!9/4.jt!hbnfpsfcb/! 

97/2/!nfpsfhwbsj-!Tfnplmfcvmj!xbljUywb;!`yvUj!dbmj!xzwjmj! 

eb!nfuj!bsbgfsj~-!bov-!nplmfe;!`yvUj!xzwjmj~/ 

_ miTxariT, romeli ufro meti raodenobaa vaSlebisa _ 

xuTi cali vaSli Tu xuTi vaSli? {toli raodenobaa} aTi 

cali mravalkuxedi Tu aTi mravalkuTxedi? 

gakeTdeba imdagvarive savarjiSoebi, rac 85.1-Si, Sedegebis 

CaweriT, SigadaSig gamoereva unaSTo SemTxvevebic, oRond erTi 

siaxliT. amgvari magaliTis dros masw.-i ityvis: _ maS 

sameulebad daTvliT miviReT ra? {xuTi cali sameuli da naS- 

Ti nuli}. mokled es ase vTqvaT: xuTi sameuli. zogjer SeiZleba 

ar vTqvaT `cali~, arc `naSTi nuli~ vTqvaT _ nulis mimateba 

gana cvlis rames? {ara} hoda `naSTi nulio~ Tu gvinda 

vityiT da davwerT, Tu ginda, ar vityviT da arc davwerT. 

bolos mTeli klasi erTad Seasrulebs 86.1-is zeda 

striqons. qveda _ saSinao davalebad. 

97/3/!UwbmtbDjop!npdfnvmpcjU;!besf!!hwjbo/!kfs!!Tfnefh/ 

jer daismis imdagvarive SekiTxvebi, rogorc 85.2-Si,Semdeg 

ganvixilavT ukve gaferadebul suraTs (86.2). _ saidan sad 

midis gza? {gogonas saxlidan tbisken} diax, gogona gamovida 

saxlidan da unda mivides tbis napiras, Semdeg ki tbaSi unda 

ibanaos. zafxulia, cxela, xom xedavT, rogor kaSkaSebs mze! 

swrafad miTxariT: mzis sxivebis raodenoba aTze metia Tu 

naklebi? {metia} kargi, gogona saxlidan gamovida. axla sad 

dgas? {gzaze, mosaxvevTan} ra aris gogonas ukan? {xeebi} 

ramdeni xea? {oTxi} amis Semdeg gogonam ras unda Cauaros? 

{Robes} Semdeg? {cxvrebs} Semdeg? {Zroxas} Zroxa gogonasTvis 

romel mxares iqneba? aba warmoidgineT Tqveni Tavi gogonas 

adgilas! {Zroxa iqneba gogonas marjvniv} Semdeg ras Cauvlis? 

{kurdRels} kurdRlis romel mxaresaa erTi xe? {ukan 

marjvniv} kurdReli gogonas romel mxares iqneba? {marcxniv}

- 238 - 

Semdeg ras Cauvlis? {naZvs} naZvi romel mxares iqneba? 

{marjvniv} sad iqneba gogona am dros? {tbis napiras} Semdeg 

ras izams? {gaixdis da ibanavebs} tbis napiras kidev ras 

xedavT? {ori yvavilia} es yvavilebi gogonas romel mxares 

iqneba? {marcxniv} gogona jer kurdRels Cauvlis Tu jer 

Robes? cxvrebTan ufro gvian gaivlis Tu ZroxasTan? yvelaze 

gvian ras Cauvlis? jer ZroxasTan Caivlis da Semdeg RobesTan 

Tu jer RobesTan da Semdeg ZroxasTan? {jer RobesTan da 

Semdeg ZroxasTan}. 

97/4/! Ubwj! 9/4.jt! Tfkbnfcb! eb! hbnfpsfcb/! ! ! ! _ radara 

mravalkuTxedebi SeiZleba davinaxoT naxazze (86.3)? 

gakeTdeba iseve, rogorc 84.4. sul eqvsi mravalkuTxedia: erTi 

xuTkuTxedi, sami samkuTxedi da ori oTxkuTxedi, 1 + 3 + 2 = 6. 

97/5/!`txsbgj!qbtvyfcj!`!fsUnbofUt~/!ix. 76.4. 

97/7 keTdeba TvalzomiT. garda amisa, davaleba kompleqsuria 

[amis Sesaxeb ix. 35.0-Si]. aseve unda damuSavdes momdevno 

gakveTilebis kompleqsuri amocanebic, Tuki maTi amoxsna 

moswavleebs gauWirdebaT. 

97/9/ sityvieradaa: 7 oTxkuTxedi da 3 xuTkuTxedi; 4 

eqvskuTxedi da 5 oTxkuTxedi; 6 texili da 7 monakveTi. 

g 87. kvgUjt!Tfnplm/!xbljUywb/ xjo!\ Tfnefh/!yb{j!){phbebe*/ 

98/2/! fsUj! dbmj! kvgUjt! Tfnplmfcvmj! xbljUywb; 1 · 5 + 2-!! 

`fsUj! dbmj! yvUfvmj! eb! obTUj! psj~-! bov-! nplmfe-! ! 5 + 2-! 

`yvUfvmj! eb! psj~ (amasTan, ara aqvs araviTari mniSvneloba 

imas, rom saerTo raodenobaa 7). _ roca vambobT `bavSvi~ _ 

erTi bavSvia Tu ramdenime? Tu mravali bavSvia? {erTia} 

vardi da ori mixaki _ sul ramdeni yvavilia? {sami: erTi 

vardi da ori mixaki}. 

gakeTdeba imdagvarive savarjiSoebi, rac 86.1-Si, Sedegebis 

CaweriT. SigadaSig gamoereva unaSTo SemTxvevebic, maTi 

Semoklebuli wakiTxviTurT, da erTi siaxliT: SigadaSig 

gamoereva erTi cali grovis magaliTebi (rvaeuli da naSTi 

ori, erTi cxraeuli da naS-Ti sami). am dros masw.-i ityvis: _

- 239 - 

maS rvaeulebad daTvliT miviReT ra? jer miTxariT grZlad, 

srulad! {erTi cali rvaeuli da naSTi sami} mokled? {erTi 

rvaeuli da naSTi sami} roca mxolod erTi cali grovaa, 

SegviZlia vTqvaT kidev ufro mokled: _ rvaeuli da naSTi 

sami. kidev ufro mokled: rvaeuli da sami. 

_ axla daxedeT naxazs (87.1). ramdenia carieli CarCo? {eqvsi}. 

SeavseT isini dawyobili wertilebiT. pirvel CarCoSi CasviT sami 

xuTeuli. naSTi risi toli iqneba? {nulisa} meore CarCoSi CasviT 

cxraeuli da rva. naSTi risi toli iqneba? {rvisa} mesame CarCoSi 

CasviT ori oTxeuli da ori. naSTi risi toli iqneba? {orisa} 

meoTxe CarCoSi CasviT aTeuli da eqvsi. naSTi risi toli iqneba? 

{eqvsisa} mexuTe CarCoSi CasviT sami aTeuli. naSTi risi toli 

iqneba? {nulisa} meeqvse CarCoSi CasviT ori aTeuli da erTi. 

naSTi risi toli iqneba? {erTisa}. 

98/3/!UwbmtbDjop!npdfnvmpcjU;!xjo \!Tfnefh/!`sb!npyeb!bnjt! 

xjo! ! Tfnefh@~! ganvixilavT mTw-s ydis suraTs. _ ramdeni 

pepelaa mindorSi? {xuTi} marcxena mxares ramdenia da 

marjvniv _ ramdeni? {marcxnivaa oTxi da marjvniv _ erTi} 

romelia yvelaze didi pepela? {marjvena} suraTis romel 

kuTxeSia is? {zeda marjvena kuTxeSi} viTom es pepela sul am 

adgilas frens, sul axlomaxloa. daimaxsovreT didi pepela. 

axla miTxariT, vin da vin Cauara am pepelas ukve (gamoikveTeba 

sityva ùkve~!)? {gogonam, biWunam, cugam, fisom, dedalma da 

mamalma} maTTvis didi pepela romel mxaresaa? {ukanaa} axla 

warmoidgineT, rom gogona da biWuna da yvela cxoveli aseve 

ganagrZobs moZraobas gzaze, didi pepela ki iqve balaxze 

dajda da isvenebs. yvelaze adre vin Caivlis did pepelasTan? 

{frinveli Caifrens} Semdeg? {bu} Semdeg? {ixvis WuWuli} (da 

ase Semdeg: goWi, baWia, ciyunia, wero, beli, nukri, kvici, 

cikani, xbo, mela, bokveri, zRarbi, da bolos ku}. bokveri 

ufro adre Cauvlis did pepelas Tu nukri? jer beli Cauvlis 

da Semdeg baWia, Tu jer baWia da Semdeg beli? (daismis kidev 

ramdenime amdagvari SekiTxva). xbos Semdeg vin Cauvlis? kvicis 

win vin Cauvlis? (daismis kidev ramdenime amdagvari SekiTxva).

- 240 - 

danarCeni SekiTxvebi daismis 87.4-Si. 

98/4/! yb{j! ){phbebe*/! zogadi cneba `xazi~ aerTianebs 

monakveTebs, texilebsa da mrude xazebs [zustad ix. zms $ 5, 

TvalsaCinod _ mTw-s geometriuli TvalsaCinoeba]. 

jer viwyebT gameorebiT: _ rogori xazebi viciT? {swori 

xazebi, texilebi, mrudebi} swor xazs meoregvarad ra 

ewodeba? {monakveTi} dafasTan gamovidnen eliso, naTela da 

guliko. erTma daxazos monakveTi anu swori xazi, meorem _ 

texili xazi, mesamem _ mrudi xazi. risgan Sedgeba texili? 

{monakveTebisagan} diax, mruds ki gamrudebuli, momrgvalebuli 

adgilebic aqvs. texils Tu SeiZleba hqondes erTi patara 

momrgvalebuli adgili? {ara, texili mxolod monakveTebisagan 

Sedgeba} ra ewodeba texilis monakveTebis SeerTebis 

wertilebs? {texilis wveroebi} naTelam ra xazi daxaza? 

{texili} Sekruli Tu gaxsnili? gulikom ra xazi daxaza? 

{mrudi} Sekruli Tu gaxsnili? 

_ monakveTic, texilica da mrudic _ yvela erTmaneTs 

hgavs erTi ramiT. Tanac, sulerTia, texili Sekrulia Tu 

gaxsnili, mrudi Sekrulia Tu gaxsnili, sulerTia. sxvadasxvanairebi 

ki arian, magram erTi ramiT hgvanan erTmaneTs. aba 

vin mixvdeba, riTi? {yvela Zalian viwroa, yvelaferze ufro 

viwro} da amitomaa, rom samives erTad erTi saxeli aqvs _ 

xazi. monakveTic xazia, texilic xazia da mrudic xazia. xazi 

ganieri ar unda iyos, yvela xazi viwroa, Zalian viwro. 

gaixseneT, wertili riT gamoirCeva? {yvelaferze pataraa, 

wvrilia} wertili yvelafriT pataraa: sigrZec Zalian patara 

aqvs da siganec! xazi ki rogoria? sigrZiTac pataraa? {ara, 

xazi SeiZleba iyos Zalian grZeli} diax, xazi grZelia, magram 

Zalian viwroa. sigane aqvs Zalian patara, wertiliviT patara 

sigane aqvs, Torem sigrZe ki SeiZleba, rom didi hqondes! 

masw.-i dafaze daxatavs patara wres. _ warmovidginoT, rom 

es wre dafaze amoZravda da gasrialda. rogori kvali darCeba 

mas? ras daxazavs dafaze? {zols}. 

masw.-i daxazavs saTanado siganis mqone mrude zols, Tan

- 241 - 

gaWimuli ceriTa da saCvenebeli TiTiT uCvenebs, rogor 

`moZraobda~ carcis wre. 

_ aqvs Tu ara am zols sigane? aqvs, ai esaa misi sigane. ra 

siganec hqonda wres, is sigane eqneba mis kvals! axla 

warmovidginoT, rom es wertili amoZravda da gasrialda 

dafaze. rogori kvali darCeba mas? ras daxazavs dafaze? 

{xazs} aqvs Tu ara xazs sigane? {ara} imitom, rom wertils 

ara aqvs sigane! amitom arc mis kvals aqvs sigane. 

bolos, msr-Si cariel adgilas (87.3) daxazaven 

sxvadasxvagvar xazebs: oTx monakveTs, sam mrudsa da amdenive 

(am sityvis gamokveTiT) texils. es unda iyos swored daxazva, 

da ara daxatva (ese igi, monakveTebi da texilebi saxazaviT 

unda daixazos). Semdeg masw.-i klass daaxurinebs msr-s da 

ikiTxavs: _ ramdeni monakveTi daxazeT? {oTxi} texili? {sami} 

mrudi? {esec sami} maS sul ramdeni xazi dagixazavT? {4 + 3 

+ 3 = 10, oRond Canaweris gareSe, zepirad, Tuki gauWirdebaT, 

gamoiyenon ricxvTa kibe}. 

98/5/!`txsbgj! qbtvyfcj!`!kfs!!Tfnefh~/ ganvixilavT kvlav 

mTw-s ydis suraTs. SekiTxvebi imdagvarivea, rogorebic iyo 

87.2-Si, oRond amoirCeva isini, romlebze pasuxic erTsityviania: 

_ bokveri ufro adre Cauvlis did pepelas Tu nukri? 

beli ufro gvian Cauvlis Tu baWia? gogonam ufro adre 

Cauara Tu fisom? cugam ufro adre Cauara Tu mamalma? xbos 

Semdeg vin Cauvlis? kvicis win vin Cauvlis? 

meore wreze emateba SekiTxvebi advil sivrciT mimarTebaTa 

Sesaxeb (goWi vis winaa, kvici vis ukanaa da sxva); mesame wreze 

_ Znel sivrciT mimarTebaTa Sesaxeb (ciyvis marcxniv vinaa, 

beli vis marjvnivaa da sxva). droiT mimarTebaTa Sesaxeb 

SekiTxvebic grZeldeba. 

g 88. ndebsj!òbTUj•/!xjo \!Tfnefh/!Tflsvmj-!!hbytojmj!yb{fcj/!! 

99/2/![ofmj!TfnUywfwfcj;!spdb!òbTUbe~!ebhwsDfcb!tbUwbmbw! 

kvgU{f! nfuj! sbpefopcb/ bnhwbsj! ndebsj! òbTUjebo~! kvgUjtb! 

eb! obnewjmj! obTUjt! hbnpzpgb/! gakeTdeba ramdenime imdagvarive 

savarjiSo, rac 87.1-Si, Sedegebis CaweriT, SigadaSig gamoereva

- 242 - 

erTi cali jufTisa an unaSTo SemTxvevebi, maTi Semoklebuli 

wakiTxviTurT. amasobaSi masw.-i dafaze daawyobs or cal 

xuTeuls da kidev rvaeuls. _ axla vin ityvis, es rgolebi 

rogoraa dawyobili? {ori cali xuTeuli da naSTi rva} 

davweroT: 2 · 5 + 8. maS ramdeneulebadaa dawyobili rgolebi? 

{xuTeulebad} daukvirdiT _ kargadaa dawyobili es rgolebi 

xuTeulebad? 

Tuki vercerTi moswavle ver mixvdeba, masw.-i mianiSnebs: _ 

aba daukvirdiT _ xuTeulebadaa dawyobili, magram... magram ra? 

{kidev erTi xuTeuli unda dagvewyo} diax, am rvaeulidan xom 

kidev erTi xuTeuli daewyoba! gamodi da daawyve! ese igi, unda 

davawyoT ase: òri xuTeuli da kidev xuTi da kidev sami~, ese 

igi, sul ramdeni xuTeuli yofila? {sami xuTeuli da kidev 

(anu: da naSTi) sami}. es imitom moxda, rom rva metia, vidre 

xuTi. amitom ar iyo kargad dawyobili! axla vin daxatavs 

aseve cudad dawyobil jvrebs? 

erTi moswavle daxatavs, jufTebs saTiTaod Semoxazavs. 

Semdeg sxva Seasworebs. 

bolos msr-Si (88.1) zeda striqonSi TviTeul mcdarad daxazul 

raodenobas gverdiTa cariel CarCoSi miuxazaven marTebulad 

daxazulebs. Semdeg qvemoT gaakeTeben saTanado Canawerebs. 

Tan TviTeuls waikiTxaven: oTxi oTxeuli da ori; sami xuTeuli. 

qveda striqoni _ saSinao davalebad {iqneba: 

6 · 2 + 1 da 7 · 2}. 

99/3/!UwbmtbDjop!npdfnvmpcjU;!xjo!!Tfnefh/!`ft!sjt!Tfnefh\ xjo! 

npyeb@~!jer ganvixilavT ganvlil suraTs 86.2. _ gogona Zroxas 

ris Semdeg Cauvlis? kurdRels ris win Cauvlis? (da a. S.) 

Semdeg ganvixilavT suraTebs 88.2. _ jambazi aTamaSebs bur- 

Tebs. jer is magidasTan mivida. romelia es pirveli suraTi? 

{marcxena} ramdeni burTia magidaze? {aTi} amis Semdeg jambazma 

ramdeni burTi aiRo? {oTxi} magidaze ramdeni darCa? {eqvsi} 

Semdeg jambazma ra qna? ramdeni burTi moaklda magidas? {ori} 

saidan ici? {magidaze darCa oTxi burTi, eqvsze oriT naklebi, 

manamde ki magidaze eqvsi burTi ido} ese igi, jambazs kidev

- 243 - 

ramdeni burTi auRia? {kidev ori} sul ramdeni burTi aaTamaSa 

haerSi jambazma? {oTxs mivumatoT ori, eqvsi} magidaze ramdeni 

burTi darCa? {oTxi} ramdeni burTia zemoT haerSi, romlebic 

suraTze ar Cans? {sami} saidan ici? {Cans haerSi sami burTi, 

danarCeni zemoTaa, eqvss gamoklebuli sami udris sams}. 

_ maS, Tavidan, pirvelad jambazma ramdeni burTi aiRo? 

{oTxi} amis Semdeg? {kidev ori, sul eqvsi}. 

99/4/!Tflsvmj!eb!hbytojmj!yb{fcj/!!hbytojmj!yb{fcjt!cpmp! 

xfsujmfcj! )cpmpfcj*/! _ ra hqvia monakveTebs, texilebsa da 

mrudebs, yvelas erTad? {xazebi} monakveTs Tu aqvs boloebi? 

{ki} ramdeni? {ori} texils? {gaxsnils aqvs, Sekruls _ ara} 

ramdeni bolo aqvs gaxsnil texils? {ori} mruds Tu aqvs 

bolo wertilebi? {gaxsnils aqvs, Sekruls _ ara} ramdeni 

bolo aqvs gaxsnil mruds? {ori} maS, rogor xazebs ara aqvs 

boloebi? {Sekrul xazebs} gaxsnil xazs ramdeni bolo aqvs? 

{ori} yvelanairs? {ki, monakveTs, texils, mruds _ yvelas orori 

aqvs bolo} ra nakvTia xazis bolo? {wertili}. 

Semdeg gadaiSleba mTw geometriul sqema-TvalsaCinoebaze. 

daaTvaliereben xazebs _ rogoraa isini dajgufebuli. _ aba 

daakvirdiT naxazebs, xazebis dajgufebas da moifiqreT _ SeiZleba 

Tu ara, rom texili arc Sekruli iyos da arc gaxsnili? 

{SeiZleba, mTw-Si mocemulia jgufi `sxvebi~}. gamovidnen 

avTo da aleko da dafaze daxazon TiTo-TiTo aseTi sxvagvari 

texili, romelic arc Sekrulia da arc gaxsnili. mrudi? yvela 

mrudi an Sekruli unda iyos da an gaxsnili? Tu sxvagvaric 

SeiZleba iyos? {SeiZleba, mTw-Si mocemulia jgufi 

`sxvebi~}. gamovidnen nona da nana da daxazon TiTo-TiTo aseTi 

sxvagvari mrudi, romelic arc Sekrulia da arc gaxsnili. 

masw.-ic daxazavs qarTuli `g~ asos magvar xazs. _ es xazi 

Sekrulia Tu gaxsnili? {arc Sekrulia da arc gaxsnili} 

diax, ai am adgilas xom Sekrulia, wresaviTaa Sekruli, magram 

aq _ gaxsnilia, ai es erTi bolo wertilic aqvs! mTlianad 

xazi ki arc Sekrulia da arc gaxsnili. es sxvagvari xazebi 

rTulia da maT pirvel klasSi aRar viswavliT. ese igi,

- 244 - 

xazebis sul ramdeni jgufia? daxedeT suraTs! (mTw-Si). 

sabolood, xazebis eqvsi jgufia: monakveTebi, Sekruli 

texilebi, gaxsnili texilebi, Sekruli mrudebi, gaxsnili 

mrudebi da sxvebi _ rTuli xazebi (arc Sekruli da arc 

gaxsnili texilebi an mrudebi). 

bolos, msr-Si (88.3) TviTeuli am jgufis xazebs sxvadasxva 

ferebiT Seaferadeben (magaliTad: monakveTebi _ Savad, 

gaxsnili texilebi _ lurjad, Sekruli texilebi _ wiTlad, 

gaxsnili mrudebi _ mwvaned da Sekruli mrudebi _ yviTlad). 

gverdiT da qvemoT, cariel adgilas, daxataven kidev TiTo- 

TiTo xazs TviTeuli am jgufidan, Sesabamisi feriT. 

99/5/!UbnbTj!`hvebTj!yfmjt!Dbzpgb!eb!hbnpdopcb~/ ix. 48.4. 

99/:/ rogorc yovelTvis, amdagvari amocanebi ricxvTa 

kibeze sruldeba. 

g 89. hspwjt!ebUwmb!tywbebtywb!kvgUfcjU-!npoblwfUj/! 

9:/2/! fsUjebjnbwf! hspwjt! ebUwmb! tywbebtywb! kvgUfcjU-! 

obTUjbo.vobTUpe/! magnitur dafaze grovaa 9 rgolisa. masw.-i 

rig-rigobiT gamoiyvans sxvadasxva moswavleebs da daawyobinebs: 

erTeulebad, wvilebad, sameulebad, oTxeulebad... cxraeulebad. 

TviTeul Sedegs dawers dafaze erTi svetSi, Tanac, orgvarad, 

grZlad da gverdiTa svetSi _ mokled: 

9 · 1 + 0 9 · 1 

4 · 2 + 1 4 · 2 + 1 

3 · 3 + 0 3 · 3 

2 · 4 + 1 2 · 4 + 1 

1 · 5 + 4 

1 · 7 + 2 1 · 7 + 2 

1 · 8 + 1 1 · 8 + 1 

1 · 9 + 0 1 · 9 

moswavleebi am Canawerebidan mokleebs (marjvena svetisas) 

msr-Sic Caweren (89.1-is cariel badeSi). Semdeg TviTeul am 

Canawers adgilidan waikiTxaven sxvadasxva moswavleni. 

_ ai, ramdeni sxvadasxva Canaweria, magram mTliani

- 245 - 

raodenoba yve-lasi tolia. risi tolia? {cxrisa} diax, cxra 

SegviZlia davTvaloT erTeulebad da miviRebT amas 

(miuTiTebs), SegviZlia davTvaloT wyvilebad da miviRebT amas 

(miuTiTebs), (da ase Semdeg). 

9:/3/! obtxbwm! espjU! njnbsUfcbUb! eblbwTjsfcb! sjhpcjU! sjdyw. 

fcUbo/! ganvixilavT suraTebs 89.2. maTze gamoxatulia mcenaris 

gaRiveba da zrda, oRond suraTebis Tanmimdevroba areulia. 

_ Tavidan Tesli miwaSia da gaRivdeba, ese igi, mas Rivi 

gamoezrdeba. romeli yofila pirveli suraTi? daawereT mas 

erTiani, zeda marcxena kuTxeSi! Semdeg ra moxda? {Rivi 

gaizarda} kidev ra moxda? {fesvebic gaizarda} daawerT am 

suraTs oriani, zeda marcxena kuTxeSi! Semdeg ra moxda? {Rivi 

miwidan amovida, ese igi, mcenare aRmocenda} ra unda davweroT 

am suraTis zeda marjvena kuTxeSi? {samiani} Semdeg ra moxda? 

{mcenares gamoezarda ylorti da ori patara foToli} ra 

unda davweroT am suraTis zeda marjvena kuTxeSi? {oTxiani} 

Semdeg ra moxda? {ylorti gaifurCqna, orive foToli gaizarda} 

fesvebi? {kidev gaizarda} ra unda davweroT am suraTis 

zeda marjvena kuTxeSi? {xuTiani} Semdeg ra moxda? {ylorti 

ufro gaifurCqna da fesvebi kidev gaizarda} ra unda davweroT 

am suraTis zeda marjvena kuTxeSi? {eqvsiani} Semdeg ra 

moxda? {gaifurCqna erTi yvavili da foTlebi gaizarda} ra 

unda davweroT am suraTis zeda marjvena kuTxeSi? {Svidiani} 

Semdeg ra moxda? {meore yvavili gaifurCna} ra unda davweroT 

am suraTis zeda marjvena kuTxeSi? {rviani}. 

_ axla miTxariT: ra moxda yvelaze adre? sul pirvelad? 

{Tesli miwaSi gaRivda} meored? mesamed? (da ase Semdeg). ra 

moxda yvelaze gvian? sul bolos? 

9:/4/! psj! xfsujmjt! TffsUfcb! tywbebtywbhwbsj! yb{fcjU/! 

npoblwfUj! `! txpsj! yb{j/! masw.-i dafaze dasvams or wertils, 

erTmaneTisagan ori-sami mtkavelis daSorebiT. rig-rigobiT 

gamoiyvans moswavleebs da am wertilebs SeaerTebinebs (sityvis 

`SeerTeba~ gamokveTiT) jer mrude xaziT, Semdeg _ texili 

xaziT, Semdeg _ sxva texili xaziT, Semdeg _ sxva mrude

- 246 - 

xaziT da bolos _ monakveTiT (sasurvelia, sxvadasxva feris 

carcebiT). _ ramdeni texiliTaa SeerTebuli es wertilebi? 

{oriT} kidev mravali sxva texiliT Tu SegviZlia maTi 

SeerTeba? {ki} ramdeni mrudiTaa SeerTebuli es wertilebi? 

{oriT} kidev mravali sxva mrudiT Tu SegviZlia maTi Seer- 

Teba? {ki} ramdeni monakveTiTaa SeerTebuli es wertilebi? 

{erTiT} kidev mravali sxva monakveTiT Tu SegviZlia maTi 

SeerTeba? {ara} diax, ori wertili mxolod erTi, erTaderTi 

monakveTiT SeiZleba SeerTdes. riTia gamorCeuli monakveTi 

sxva xazebisagan? rogori xazia igi? {swori xazia}. 

sakiTxi sityvisa èrTaderTi~ da monakveTis erTaderTobisa 

aRar gaRrmavdeba. 

Semdeg moswavleebi msr-Si (89.3) dasmul or wertils 

SeaerTeben sxvadasxvagvari xazebiT (rogorc dafaze), 

gamoiyeneben ra sxvadasxva feris fanqrebs. 

9:/5/! `sjdywUb! ebnxlsjwfcb~/ igivea, rac 49.4, oRond 

moswavleebs urigdeba moZravi cifrebi 0-dan 9-mde. 

9:/7/ amgvari amocana pirveladaa, Semdeg ki yoveldRiurad 

iqneba. saWiroa axsna-ganmarteba da Tan Cveneba, Tu rogor unda 

imoqmedon Sin. yvelafers, rac saWiroa anu rac pirobaSi 

weria, gaakeTeben klasSi, oRond Sedegebs ar Caweren. masw.-i 

ubralod ityvis: Sin am dros CawerT `sami cali wyvili~, da 

ase Semdeg. moswavleebi Sin gaimeoreben imave moqmedebebs da 

Tan Sedegebs Caweren. 

g 90. hbnfpsfcb/!!kbnfcjt!Tfebsfcb/!tjhs[jt!Uwjtfcb/! 

:1/2/! Ubwj! 22/2.jt! Tfkbnfcb! eb! hbnfpsfcb/! ! gakeTdeba iseTive 

savarjiSo, rogorc 89.1-Si, oRond cxriani grovis nacvlad 

iqneba rviani grova. dafaze daweril TviTeul gamosaxulebas 

waikiTxaven grZladac da mokledac. Semdeg mokleebs Caweren 

msr-Si (90.1). 

:1/3/! kbnfcjt! Tfebsfcb-! spdb! qjswfmj! Tftblsfcjb! fsUjeb. 

jhjwf/!!!!dafaze daiwereba erTmaneTisgan cota moSorebiT: 

3 + 6 3 + 4.

- 247 - 

_ ramdeni jami weria dafaze? {ori} romelia pirveli 

jamis pirveli Sesakrebi? {3} meore Sesakrebi? {6} meore jamis 

pirveli Sesakrebi? {3} meore Sesakrebi? {4} romeli 

Sesakrebebi aqvT erTidaigive am jamebs? {pirveli} axla 

SeadareT _ romeli jamia meti? saidan ici? 

Tuki moswavlem iangariSa da ise Seadara, masw.-i ikiTxavs: _ 

saWiro iyo angariSi? angariSis gareSe gana ver mivxvdebodiT, 

romelia meti? sams rom eqvsi mivumatoT da sams rom oTxi 

mivumatoT _ romeli iqneba meti? Tuki gaigeT, axla me sxva 

jamebs davwer da Tqven swrafad SeadareT, nu iangariSebT, ise: 

6 + 4 da 6 + 5 (ara aqvs mniSvneloba imas, rom 11 jer 

ar gviswavlebia). 

_ axla rveulSi vnaxoT jamebi (90.2). ramdeni jami weria? 

{eqvsi} swrafad SeadareT pirveli da meore, CawereT metobis 

an naklebobis saWiro niSani (da ase Semdeg). 

bolos, masw.-i sxvadasxva moswavleebs dausvams amgvar 

SekiTxvebs: _ romelia mesame jamis meore Sesakrebi? (da 

sxva). 

:1/4/!tjhs[ffc{f;!!Uvlj!b > j!!eb!!j > p-!!nbTjo!!b > p/ 

esaa sigrZis `tranzitulobis~ Tviseba. jer gavameorebinebT 

simetriulobas (82.3 da 83.3), umjobesia warmodgeniT. masw.-i 

dawers dafaze: j < f f > j. 

_ axla vnaxoT sxva rame, romelia ufro grZeli _ Toki 

Tu saxazavi. gamovides levani da Seamowmos! (daatolebs). kargi, 

davweroT: T > s. axla gamovides daTo da Seadaros 

saxazavisa da am monakveTis sigrZeebi. kargi, amasac davwer: 

s > m. irakli, romeli yofila ufro grZeli _ Toki Tu 

saxazavi? {Toki} saxazavi Tu monakveTi? {saxazavi} axla vin 

gvetyvis dautoleblad, vin mixvdeba, Tokia ufro grZeli Tu 

monakveTi? {Toki} ratom? {Toki xom saxazavze grZelia, 

saxazavi Tokze moklea, monakveTi ki kidev ufro moklea}. 

diax, esec davweroT. 

sabolood dafaze amgvari Canaweri iqneba, erT striqonSi: 

T > s s > m T > m

- 248 - 

Semdeg masw.-i klass warmodgeniT Seadarebinebs sigrZeebs: _ 

warmoidgineT, rom texili ufro grZelia, vidre monakveTi, 

monakveTi ki ufro grZelia, vidre mrudi. aba, jer gaimeoreT, 

romeli romelze grZelia! (gaimeoreben). axla moifiqreT _ 

romeli iqneba ufro grZeli _ texili Tu mrudi? ratom? 

_ axla warmoidgineT, rom marTkuTxedi ufro grZelia, 

vidre texili, texili ki ufro grZelia, vidre monakveTi (da 

kvlav gavameorebinebT). romeli iqneba ufro grZeli _ 

marTkuTxedi Tu monakveTi? ratom? 

TfojTwob/ masw.-ma unda icodes: amgvar amocanebSi, roca 

monacemebi araa TvalsaCinod mocemuli, mTavari siZnele isaa, 

rom bavSvs ar daaviwydes es monacemebi _ romeli romelze 

grZelia. bavSvs orive monacemi erTdroulad unda hqondes 

aqtiur mexsierebaSi. ese igi, amgvar amocanebSic mTavaria 

pirobis kargad gaazreba. amitom saWiroa misi gameoreba da 

gonebaSi kargad warmodgena. 

:1/5/! `txsbgj! qbtvyfcj~! _! `.jU! nfu.oblmfcpcb~/ ix. 51.4, 

oRond axla viwyebT òriT metiT~. 

:1/7/ masw.-i ityvis, rom es jamebi iseve unda Seadaron, 

rogorc klasSi _ ukeTesia, angariSis gareSe. 

------------------------------------------------------------------------------------ 

I-12. Uwmb! bUfvmfcjU- magram es iqneba jufTebiT Tvlis 

mxolod erTi kerZo SemTxveva. arsebiTad axali araferi araa. 

TvalsaCinod mocemul raodenobaSi gamovyofT aTeulebs. amis 

safuZvelze vaswavliT ricxvebs aTidan ocamde, jer mxolod 

sityvierad, maT saxelwodebaTa Tanmimdevruli gaazrebiT (cifrul 

CanawerTa gareSe!). viwyebT {fqjs! Tflsfcb.hbnplmfcbt! pdjt! 

gbshmfcTj dominoseburi TvalsaCinoebiT. 

erTidan aTamde ricxvebis Semotanisas saguldagulod 

vmuSaobdiT nivTTa sxvadasxva grovebze. axla es aRaraa saWiro, 

radganac klass ukve mtkiced eqneba SeTvisebuli is, rom 

raodenoba ar icvleba grovis ganlagebis SecvliT. ricxvTa 

Sedarebac ufro swrafad SeiZleba, aRaraa saWiro dawyvilebani. 

arsebiTia, rom am droisaTvis moswavleebma ukve kargad ician

- 249 - 

`-iT metoba~ da `-iT nakleboba~ meore xaziT. 

Sekreba-gamokleba àTeulze gadausvlelad~, rac ar Cans 

dominosebur TvalsaCinoebaze (11+2, 18 – 3 da misT.), jer ar 

ganixileba. 

Cveulebriv, Sekreba-gamokleba àTeulze gadasvliT~ iswavleba 

erTi Sesakrebis 10-mde Sevsebis xerxiT, anu nawil-nawil; 

Sesabamisad gamoklebac nawil-nawil gamoklebis xerxiT iswavleba. 

magram sinamdvileSi es Zalian Zneli gzaa. moswavlem gonebaSi 

mravali moqmedeba unda Seasrulos. dawvrilebiT CavweroT TiTo 

magaliTi: 

7 + 8 = 7 + (10 – 7) + (8 – (10 – 7)) = 

= 7 + 3 + (8 – 3) = 10 + 5 = 15. 

16 – 9 = 16 – 6 – (9 – 6) = 10 – (9 – 6) = 10 – 3 = 7. 

arsebiTad, amdeni nabiji unda gaiazros moswavlem gonebaSi. 

pirvelklaselisTvis es Zalian Znelia. amitomacaa, rom am xerxs 

mxolod maTematikuri monacemebis mqone moswavleebi euflebian. 

da maTTvisac ki es xerxi aris, Tumc kargi da sasargeblo, magram 

mainc damatebiTi amocana. igi ar aadvilebs Sekreba-gamoklebas, 

piriqiT. TviT Sekreba-gamokleba ufro advilia, vidre am xerxis 

gonebaSi gamoyeneba. 

yovelive amis gamo Cven Sekreba-gamoklebas àTeulze gadasvliT~ 

vaswavliT Cveni mTavari xerxiT dominoseburi TvalsaCinoebiT. 

nawil-nawil Sekreba-gamoklebis xerxs Cven ar viyenebT. 

amasTan, nawil-nawil Sekreba-gamokleba kargi algebruli amocanaa. 

amitom Cven mas mainc vaswavliT, oRond ara Sekreba-gamoklebisaTvis, 

aramed sxva, algebris TemaSi, meore xaziT, Tanac, 

meore klasSi. amiT kidev erTxel vimeorebT da vaviTarebT 20-is 

farglebSi Sekreba-gamoklebas. oRond masw.-ma unda gaiTvaliswinos, 

rom am xerxs moswavleTa nawili mainc ver aiTvisebs. 

sazogadod, 20-is farglebSi Sekreba-gamokleba pirvel klasSi 

isedac ver iqneba myarad aTvisebuli. amaze meore klasSi isedac 

mogviwevs bevri muSaoba. 20-is farglebSi Sekreba-gamoklebis myari 

da mtkice dauflebis gareSe azri ara aqvs ariTmetikis xaziT 

winsvlas.

- 250 - 

----------------------------------------------------------------------------------- 

g 91. bUfvmUb!hbnpzpgb!{fqjsbe/!kbnfcjt!Tfebsfcb/! 

:2/2/! èpnjoptfcvsj~! bUfvmfcjU! ebybuvm! sbpefopcbTj! 

bUfvmUb! hbnpzpgb! {fqjsbe-! DbobxfsUb! hbsfTf; `psj! dbmj! 

bUfvmj~!bov!`psj!bUfvmj~-!`swb!dbmj!bUfvmj~-!`yvUj!bUfvmj! 

eb! obTUj! psj~-! `fsUj! bUfvmj! eb! obTUj! fsUj~! eb! tywb/!! 

Tftbcbnjtj! trfnfcjt! ebybuwb/! ! rogorc yovelTvis, viwyebT 

nivTebiT. dafazea 30-40 cali rgolis grova da romelime 

moswavle mas aTeulebiT daawyobs da daTvlis. Sedegi zepirad 

warmoiTqmis: _ sami cali aTeuli da naSTi oTxi. anu mokled: 

sami aTeuli da oTxi. 

Semdeg masw.-i klass gaaxsenebinebs, `lamazad~ rogor 

ixateba aTeuli (`dominoseburad~). didad daixateba dafaze. _ 

SexedeT, ori risgan Sedgeba aTeuli? {ori xuTeulisgan} 

gareTa CarCo romel nakvTs mogagonebT? {marTkuTxeds} amitom 

aTeuls mokled aRvniSnavT imdagvarive marTkuTxa CarCoTi, ai 

ase, mravali wertili rom aRar davsvaT. davimaxsovroT, rom 

carieli CarCo _ aTeuls aRniSnavs, viTom SigniT wertilebi 

aqvs. axla rogor SeiZleba CavweroT es Cveni grova 

aTeulebiT? (daixateba sami CarCo da kidev oTxeuli). ai es 

_ erTi aTeuli, es _ meore aTeuli, es _ mesame aTeuli da 

esec naSTi _ oTxeuli. 

Semdeg gaakeTeben imaves, oRond msr-Si (91.1). waikiTxaven, 

ramdeni aTeulia, ramdenia naSTi da qvemoT Sesabamis sqemebs 

daxataven. 

:2/3/!kbnfcjt!Tfebsfcb-!spdb!nfpsf!Tftblsfcjb!fsUjebjhjwf/ 

yvelaferi gakeTdeba iseve, rogorc 90.2-Si, oRond 

erTidaigivea meore Sesakrebebi. 

:2/4/!tjhs[ffc{f;!!Uvlj!j < b!!eb!!b < p-!!nbTjo!!j < p/!! 

pirveli samuSao (datolebani) gakeTdeba iseve, rogorc 90.3- 

Si, oRond metoba Seicvleba naklebobiT, ùfro grZelia~ _ 

ùfro mokleTi~. 

warmodgeniT SedarebaSi ki mcireodeni siaxleebia: _ axla 

warmoidgineT, rom texili ufro moklea, vidre mrudi, mrudi

- 251 - 

ki ufro moklea, vidre monakveTi. am sam xazTagan romelia yvelaze 

mokle? {texili} yvelaze grZeli? {monakveTi} ratom? 

_ warmoidgineT, rom kvadrati ufro moklea, vidre 

marTkuTxedi, samkuTxedi ki ufro grZelia, vidre marTkuTxedi. 

am sam nakvTagan romelia yvelaze grZeli? 

{samkuTxedi} yvelaze mokle? {kvadrati} ratom? 

:2/5/!`txsbgj!qbtvyfcj~!`!bohbsjTj/!SekiTxvebia Cveulebrivi 


Semdeg _ aTis farglebSi. 

g 92. bUfvmjt!hbnpzpgb/!psj!sbnjt!tjhs[jt!Tfebsfcb/ 

:3/1. 91.9-Si saWiroa dasabuTeba {sawoli da oTaxi rom 

davatoloT, sawolis Tavi unda mivdgaT kedelTan (dafaze 

daixazeba sqema). oTaxi ufro grZelia, radganac sawoli 

oTaxSia, Sigaa}. 

:3/2/! UfsUnfujbo! hspwbTj! bUfvmjt! hbnpzpgb/! kvgUfcjU! 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!fsUj~/!nplmfe;!`bUfvmj! 

eb! fsUj~-! bov;! `bUj! eb! fsUj~/! `sbnefojU! nfujb! bU{f@~! `! 

`fsUjU! nfuj~/! hbb{sfcb! tjuzwjtb! Ù.fsU.nfuj~/! UfsUnfujt! 

Tfebsfcb!obtxbwm!sjdywfcUbo/!!!dafazea TerTmetiani grova. _ 

gamodi da daTvale aTeulebiT! ramdenia? {erTi cali aTeuli 

da naSTi erTi} mokled rogor vTqvaT? {aTeuli da erTi} 

diax, aTeuli da erTi. es raodenoba ramdeniTaa meti, vidre 

aTi? {erTiT metia} saidan ici? {imitom, rom aTeulis garda 

rCeba erTi} diax, erTiT meti. axla daukvirdiT am sityvebs, 

kargad moismineT: aTze erTiT meti, ufro mokled: aT-erTmeti, 

TerTmeti. TerTmeti. am raodenobas hqvia TerTmeti. 

ratom hqvia ase? {imitom, rom aTze erTiT metia} risi toli 

yofila TerTmeti? {aTs mimatebuli erTi} lamazad rogor 

SeiZleba davxatoT TerTmeteuli? 

gamoiTqmeba sxvadasxva varaudi, moswavleebi dafasTan 

gamovlen da daxataven. Tan sxvebi daTvliT Seamowmeben, 

marTla TerTme-teulia daxatuli Tu ara. saWiroa, rom xuTmaeqvsma 

moswavlem mainc daxatos, raTa gamoCndes, Tu rogor

- 252 - 

sxvadasxvagvarad SeiZleba erTidaimave TerTmeteulis dawyoba. 

Tuki arcerTma moswavlem ar daxata ise, masw.-i SesTavazebs: _ 

TerTmeti tolia aTisa da erTisa da am ricxvs saxelic ase 

daarqves. amitom modi, Cvenc ase davxatoT: aTi da erTi 

(daxatavs `dominoseburad~ _ rogorcaa mTw-Si, Tumca, TviT 

mTw am gakveTilze saWiro araa). 

_ romelia ufro meti _ aTi Tu TerTmeti? {TerTmeti} 

ramdeniT? {erTiT} cxra ufro metia Tu TerTmeti? 

{TerTmeti} saidan ici? {TerTmeti metia, vidre aTi, aTi ki 

metia, vidre cxra} romelia naklebi _ Svidi Tu TerTmeti? 

{Svidi} saidan ici? (da ase Semdeg _ kidev oriode Sedareba). 

Cvens mier naswavl ricxvebs Soris romelia yvelaze patara? 

{nuli} yvelaze didi? {TerTmeti}. 

_ daiTvale nulidan TerTmetamde! Sen ki ukuRma daiTvale 

TerTmetidan nulamde. 

:3/3/! tywbpcbUb! Tfebsfcb-! spdb! tblmfcjb! fsUjebjhjwf/!! 

yvelaferi gakeTdeba iseve, rogorc 90.2-Si, oRond jamebi 

Seicvleba sxvaobebiT, erTidaigive ki saklebebia. sxva 

yvelaferi iseve: gairkveva, romelia mesame sxvaobis saklebi (an 

maklebi) da ase Semdeg, Sesruldeba msr-s 92.2 da sxva. 

:3/4/! psj! sbnjt! tjhs[jt! Tfebsfcb! Upljt! tbTvbmfcjU/! ! _ 

rogor SevadaroT erTmaneTs fanjris sigane da karis sigane? 

rom ver davatolebT? 

Tuki ver mixvdebian, masw.-i odnavi miniSnebiT miaxvedrebs, 

rom Toki unda gamoviyenoT. jer TokiT gazomaven karis siganes 

(Tokis gadakecviT saTanado adgilas). Tan dafaze daiwereba: 

k = T. Semdeg imave Toks daazomeben fanjaras, dafaze 

gverdiTve daiwereba esec: T < fF. bolos gairkvevea, f 

ufro metia Tu k (rogorc 91.3-Si). 

k = T T < Ff k < f 

aseve Sedardeba dafaze daxazuli monakveTi karis siganes 

an rafas, dafaze daxazuli maRali marTkuTxedis simaRle 

merxis siganes da kidev erTi-ori sxva ram. 

:3/5/!`txsbgj!qbtvyfcj~!`!oblwUfcjt!dopcb/ ix. 47.4.

- 253 - 

:3/9/ saWiroa yuradRebis gamaxvileba: _ ramdeni jami 

weria? {oTxi} ra aqvs erTidaigive oTxive jams? {meore 

Sesakrebi} pirveli Sesakrebi ki rogor icvleba? {erTiT 

matulobs} (meti arafris Tqma araa saWiro, vinc Sin ver 

gaakeTebs, momdevno gakveTilze gaigebs). 

g 93. tywbpcbUb!Tfebsfcb-!nsvef!yb{jt!hb{pnwb!UpljU/! 

:4/2/! {fqjsj! Tflsfcb;! 10+1, 1+10, 9+2, 2+9, ... , 6+5, 5+6.!!!! 

mTw gadaiSleba `11~-ze. _ TerTmeti wre am jamSi rogor 

aris dawyobili? {masw.-i uTiTebs aT da erT wreebs). 

maSasadame, ramdenia aTs mivumatoT erTi? {TerTmeti} (Semdeg 

masw.-is miTiTebiT aTvaliereben jams, romelSic wreebi 

dawyobilia cxrad da orad. aTvaliereben marcxnidanac da marjvnidanac). 

am jamSi ramdeni wrea? {TerTmeti} rogoraa dawyobili 

wreebi? {cxra da ori} ramdenia cxras mivumatoT ori? 

ors mivumatoT cxra? {TerTmeti} axla davakvirdeT am jams. 

ramdeni wrea? {11} TerTmeti wre rogor aris dawyobili? 

{dawyobilia rvad da samad} ras mixvdiT? am jamis mixedviT 

vin ityvis Sekrebis or magaliTs? {8+3=11 da 3+8=11} diax, 

rvas mivumatoT sami, sams mivumatoT rva _ TerTmetia. 

aseve saTiTaod gairCeva Sekrebis yvela danarCeni magaliTi. 

moswavleebi rig-rigobiT iTvlian wreebs jer marcxnidan, mere 

_ marjvnidanac, ris Semdegac srulad amboben Sekrebis or 

magaliTs. 

_ vin romeli jami daimaxsovra? (moswavleebi zepirad 

amboben Sekrebis magaliTebs). 

:4/3/!tywbpcbUb!Tfebsfcb-!spdb!nblmfcjb!fsUjebjhjwf/ 


:4/4/!nsvef!yb{jt!hb{pnwb!Upljt!tbTvbmfcjU/!njtj!Tfebsfcb! 

txps!yb{Ubo/! jer gavimeorebT erT imdagvarive savarjiSos, 

rogoric iyo 92.3-Si. 

Semdeg masw.-i dafaze daxazavs did mrude xazs, ramdenime 

klakniliT. _ es ra nakvTia? {xazia} rogori xazia? {mrude} 

rogor gavigoT, es mrudia ufro grZeli Tu am magidis es

- 254 - 

kide? (mrudi iseTi sigrZisa unda iyos, rom Zalian advilad 

ar Candes TvalzomiT). 

gamoiTqmeba sxvadasxva azri. albaT, wina safexuris gavleniT, 

ityvian, rom TokiT gavzomoT magidaca da mrudico. _ magram es 

mrudi xazi rogor unda gavzomoT TokiT? {unda gavayoloT!}. 

gamova moswavle da gazomavs (ramdenime moswavlis 

daxmareba dasWirdeba, raTa Toki ar Camousrialdes!); Semdeg 

gazomavs magidasac da masw.-i Sedegebs dafaze dawers iseve, 

rogorc 92.3-Si. 

_ saxazaviT Tu SevZlebdiT magidisa da am mrudis 

Sedarebas? {vera} ratom? {imitom, rom saxazavi ar 

moiRuneboda da mruds ver gavayolebdiT}. 

gakeTdeba kidev erTi amdagvari savarjiSo: erTmaneTs Sedardeba 

dafaze daxazuli texilisa da, magaliTad, fanjris 

sigrZeebi. 

:4/5/! `txsbgj! qbtvyfcj! `! lblvo.ubTfcj~/ ix. 18.4, oRond 

raodenoba aRwevs 11-mde. 

:4/9-Si rig-rigobiTa samkuTxedi da oTxkuTxedi. 

g 94. kbnfcjt!Tfebsfcb/!npoblwfUj!`!zwfmb{f!nplmf!yb{j/ 

:5/1/ 93.8-Si saWiroa kanonzomierebis sityvieri Camoyalibebac: 

_ nakvTebis rigi aseTia: samkuTxedi, oTxkuTxedi, 

samkuTxedi, oTxkuTxedi, samkuTxedi, oTxkuTxedi... sul asea, 

rig-rigobiT, samkuTxedi da oTxkuTxedi. 

:5/2/!{fqjsj!hbnplmfcb;!22`2-!22`3-!///!-!22`21/!mTw gadaiSleba 

`11~-ze. _ am jamSi rogoraa dawyobili TerTmeti wre? {aTad 

da erTad}. maSasadame, aTisa da erTis saerTo raodenobas ra 

hqvia? {TerTmeti} risi tolia TerTmets gamovakloT erTi? 

{aTisa} TerTmets gamovakloT aTi? {erTisa} srulad gaimeoros 

TinaTinma! axla am jamSi TerTmeti rogoraa dawyobili? {cxrad 

da orad} risi tolia TerTmets gamovakloT ori? ... da ase 

Semdeg _ saTiTaod gairCeva TviTeuli `jami~ (anu elifsi). 

gamoklebis TiTo magaliTs srulad gaimeorebs TiTo moswavle. 

_ vin romeli sxvaoba daimaxsovra? (moswavleebi zepirad

- 255 - 

amboben gamoklebis magaliTebs). 

:5/3/! kbnfcjt! Tfebsfcb-! spdb! fsUfsUj! Tftblsfcjb! fsUjebj. 

hjwf/!!!jer gameoreba _ dafaze daiwereba Sesadareblad: 

4 + 6 4 + 7 

Sedardeba iseve, rogorc 90.2-Si. 

dafaze daiwereba Sesadareblad: 5 + 6 6 + 4 

aq erTi siaxlea: saWiroa yuradRebis gamaxvileba imaze, 

rom 5+6 tolia 6+5 -isa, ris Semdegac Sedareba ukve advili 

xdeba. rasakvirvelia, amas Tavad moswavleebi unda mixvdnen. 

dafaze daiwereba Sesadareblad: 5 + 4 6 + 5 

aqac winaswar SesakrebTa gadanacvlebaa saWiro. 

Semdeg Sesruldeba msr-s 94.2. TviTeul SemTxvevaSi 

pasuxi dasabuTdeba. 

:5/4/! psj! xfsujmjt! TfnbfsUfcfmj! tywbebtywbhwbsj! yb{fcjt! 

tjhs[fUb!Tfebsfcb/!npoblwfUj!`!zwfmb{f!nplmf!yb{j/ 

_ gamovides nika da dafaze dasvas ori wertili, 

erTmaneTisagan Sors. axla ciram SeaerTos es wertilebi 

mrude xaziT! axla naTiam SeaerTos isini texili xaziT! 

(umjobesia, Tuki sxvadasxva xazebi sxvadasxva feris carcebiT 

daixazeba). axla giorgim SeaerTos es wertilebi monakveTiT 

(unda gamoiyenos saklaso saxazavi _ masw.-i daexmareba). sul 

ramdeni xazi davxazeT? {sami} axla gavzomoT TviTeuli 

maTgani. riT gavzomoT? {TokiT}. 

gaizomeba iseve, rogorc 93.3-Si da Sedegebi dafaze 

daiwereba: t > m, m > s (monakveTi _ swori xazi _ 

aRvniSnoT asoTi s, xolo m iyos mrudi), t > s. 

_ romeli yofila yvelaze mokle xazi? {monakveTi}. 

amdagvarive savarjiSo Sesruldeba kidev erTxel. kvlav 

monakveTi aRmoCndeba yvelaze mokle. _ winaze romeli iyo 

yvelaze mokle? {maSinac monakveTi} diax, yovelTvis asea: or 

wertils rom sxvadasxva xazebiT vaerTebT, maTgan yvelaze 

mokle monakveTi iqneba. imitom, rom monakveTi swori xazia. 

maS, ori wertilis SemaerTebel xazTagan romelia yvelaze 

mokle? gaimeore srulad!

- 256 - 

_ axla daakvirdiT am xazebs (94.3). radara xazebia? 

{monakveTi, texili da mrudi} maTi boloebi erTadaa Tu ara? 

{ki, or wertils aerTebs} mrudi gaxsnilia Tu Sekruli? 

{gaxsnilia} texili? {igic gaxsnilia} wiTeli xazi rogoria _ 

texili Tu mrudi? {mrudi} texili ra ferisaa? {Savi} 

monakveTi? {Savi}. 

romeli iqneba maTgan yvelaze mokle? {monakveTi} saWiroa 

Tu ara gazomva, raTa es SevamowmoT? {ara, Cven xom viswavleT, 

rom monakveTi yovelTvis yvelaze moklea}. 

Semdeg dafaze daixazeba 94.3-is magvari naxazi, oRond 

mrudi ver swvdeba im or wertils, igi ufro moklea. _ am 

xazebidan romelia yvelaze mokle? 

mosalodnelia pasuxi `monakveTi~, magram masw.-i klass 

gaamaxvilebinebs yuradRebas: _ daakvirdiT, mrudi imave or 

wertils aerTebs? {ara} aba saidan viciT, rom monakveTi masze 

moklea? Cven ra viswavleT? vin ityvis zustad? maSasadame, 

gazomvis gareSe ver vityviT, es monakveTia ufro mokle Tu 

mrudi! texilisa da monakveTis sigrZeTa Sedareba Tu 

SegviZlia gazomvis gareSe? {ki, texili ufro grZelia} saidan 

ici? {texili im or wertils aerTebs} kargi, axla es mrudi 

ase rom yofiliyo? (masw.-i mruds daagrZelebs ise, rom im or 

wertilamde miaerTebs) {axla ki namdvilad viciT, rom 

monakveTi ufro moklea, vidre mrudi} ase rom iyos? (masw.-i 

mruds kidev ufro metad daagrZelebs) {axla mrudi kidev 

ufro grZelia, amitom monakveTi namdvilad ufro moklea}. 

:5/5/!Tfkjcsj!tjnbSmjt!njyfewjU!ebehpnbTj _ ix. 5.4. 

g 95. bUfvmjt!hbnpzpgb/ kbnfcjtb!eb!tywbpcfcjt!Tfebsfcb/! 

:6/1/ 94.7-Si saWiroa kanonzomierebis sityvieri Camoyalibebac: 

_ sul jamebia, romeli Sesakrebia erTidaigive? {pirveli, 

yvelaa 2} meore Sesakrebi ki rogor icvleba? {0, 1, 2, 3... 

ese igi, Semdeg unda iyos 4, 5 da ase Semdeg}. 

:6/2/! Upsnfujbo! hspwbTj! bUfvmjt! hbnpzpgb/ kvgUfcjU! 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!psj~/!nplmfe;!`bUfvmj!

- 257 - 

eb!psj~-!bov;!`bUj!eb!psj~/!`sbnefojU!nfujb!bU{f@~!`!`psjU! 

nfuj~/! hbb{sfcb! tjuzwjtb! Ù.ps.nfuj~/! Upsnfujt! Tfebsfcb! 

obtxbwm!sjdywfcUbo/ 


TormetisaTvis. 

:6/3/! kbnfcjtb! eb! tywbpcfcjt! Tfebsfcb-! spdb! fsUfsUj! 

lpnqpofoujb!fsUjebjhjwf. dafaze daiwereba Sesadareblad: 

9 – 5 3 + 9 

Sedardeba iseve, rogorc 94.2-Si, oRond masw.-i klass sxva 

siaxleze miaqcevinebs yuradRebas: _ romeli ricxvi aqvs 

erTidaigive am sxvaobasa da am jams? {9} maS romelia meti? 

{3+9} ratom? {aq cxras raRac emateba, aq ki _ akldeba} diax, 

3+9 = 9+3, da 9+3 ki metia, vidre 9-s gamoklebuli rame. 

Semdeg Sesruldeba msr-s 95.2. TviTeul SemTxvevaSi 

pasuxi dasabuTdeba. 

:6/4/!tjhs[ffc{f;!!Uvlj!!j < b!!eb!!b = p-!!nbTjo!!j < p/ 

arsebiTad, es sakiTxi ukve naswavlia. jer gavameorebinebT 

sigrZis amdavarsave naswavl Tvisebas: _ warmoidgineT, rom 

marTkuTxedi ufro grZelia, vidre samkuTxedi da xazi ki 

ufro moklea, vidre samkuTxedi. aba, gavimeoreoT, romeli 

romelze ufro grZelia! (gaimeoreben!) axla moifiqreT _ 

romelia maTgan yvelaze grZeli? {marTkuTxedi} yvelaze 

mokle? {xazi}. 

dafaze daiwereba: m > s x < s m x 

(masw.-i xmamaRla ambobs `marTkuTxedi~ da wers asos m, ambobs 

`samkuTxedi~ da wers asos s, Semdeg ikiTxavs _ aq metoba 

Cavwero Tu nakleboba? _ da ase danarCeni utolobanic). 

_ ra niSani unda CavweroT aq, cariel adgilas? {metobis} 

CavweroT! axla warmoidgineT sxva rame. mrudi ufro moklea, 

vidre texili da texilis sigrZe ki tolia Tokis sigrZisa. 

amasac gavameorebinebT, dafaze daiwereba: m < t t = T 

_ romelia ufro grZeli _ Toki Tu mrudi? {Toki} 

ratom? {Tokis sigrZe tolia texilis sigrZisa, mrudi ki 

amaze moklea}. (dafaze imave striqonSi daiwereba m < T).

- 258 - 

Semdeg msr-Si davubrundebiT 94.3-s. naxazis gverdiT 

carieli adgilia. iq cal-calke amovaxazvinebT monakveTs, 

texilsa da mruds. _ am sami xazidan romelia yvelaze mokle? 

{monakveTi} kargi. axla Savi monakveTis qvemoT daxazeT 

wiTeli monakveTi, romelic misi toli iqneba! wiTeli 

monakveTis sigrZe igive unda iyos, rac Savi monakveTisa. 

davweroT es dafaze: w = S (`wiTeli tolia Savisa~). 

davweroT isic, rom S < t (`Savi monakveTi moklea, vidre 

texili~). romelia ufro mokle _ wiTeli monakveTi Tu 

texili? {wiTeli monakveTi} ratom? {wiTelis sigrZe igivea, 

rac Savisa, texili ki amaze grZelia} davweroT: w < t. 

:6/5/! `txsbgj! qbtvyfcj! `! sbpefopcbUb! Tfwtfcb~/ ix. 39.4, 

oRond pirvel wreze raodenobas avseben aTamde, meore wreze 

da momdevnoebze _ TerTmetamde. 

:6/7/ masw.-i ityvis, rom umjobesia, Sedardes angariSis 

gareSe. momdevno gakveTilze ki moiTxovs TviTeuli utolobis 

dasabuTebas. 

g 96. Ubwj!21/3.jt!hbnfpsfcb/!oblwUfcjt!ebkhvgfcb/! 

:7/1. 95.7 da 95.8-Si saWiroa kanonzomierebaTa sityvieri 

Camoyalibebac: _ texili, mrudi, texili, mrudi, sul ase rigrigobiTaa 

texili da mrudi... 

95.8-Si sul sxvaobebia, ra aris erTidaigive? {saklebebi, 

yvelaa 9} maklebebi ki rogor icvleba? {0, 1, 2, 3... ese igi, 

Semdeg unda iyos 4, 5 da ase Semdeg}. 

:7/2/! {fqjsj! Tflsfcb: 10+2, 2+10, 9+3, 3+9, ... , 6+6.! 

yvelaferi gakeTdeba iseve, rogorc 93.1-Si, oRond TormetisaTvis. 

:7/3/! Ubwj! 21/3.jt! hbnfpsfcb! eb! Tfkbnfcb/! jer Sesruldeba 

amocana 96.2 droiT mimarTebaTa Sesaxeb: _ aq sul xuTi 

suraTia. davakvirdeT, ra xdeba suraTebze. didi kaTxidan 

patara WiqebSi asxamen yurZnis wvens. suraTebze daxatulia is, 

rac moxda erTmaneTis Semdeg. vin mixvdeba, yvelaze adre ra 

moxda? sul pirveli romeli suraTia? {pirveli marcxenaa}

- 259 - 

saidan iciT? {pirvel suraTze cotaa gadmosxmuli _ erTi 

Wiqac ki ver aivso, kaTxa ki TiTqmis savsea. amis Semdeg 

pirveli Wiqa piramde aivso, kaTxas cota moaklda. Semdeg ki 

meore WiqaSi cota Caasxes, kaTxas kidev moaklda}. 

_ moZebneT pirveli suraTi! misi romeli kuTxea carieli? 

{zeda marcxena} daawereT iq patara 1-iani! meore suraTsac 

cariel kuTxeSi daawereT patara 2-iani! (da ase Semdeg _ 

mexuTemde). _ meoTxe da mexuTe suraTebSi ki yurZnis wveni 

araa gaferadebuli _ Tqven unda gaaferadoT, kanonzomierad. 

moifiqreT, ra moxdeboda meoTxe suraTze! {meore Wiqa piramde 

aivseboda, kaTxas kidev cota moakldeboda} gaaferadeT ase! 

daukvirdiT, ar SegeSaloT! 

aseve gaaferadeben mexuTe suraTsac. _ yvelaze gvian ara 

moxda? 

bolos, gakeTdeba iseTive savarjiSo, rogoric iyo 95.2-Si. 

:7/4/! UwbmtbDjope! npdfnvmj! oblwUfcjt! ebkhvgfcb;! 

xfsujmfcj-! yb{fcj-! nsbwbmlvUyfefcj-! xsffcj! eb! tywfcj/! 

arsebiTad, es sakiTxi ukve naswavlia. gadaiSleba mTw 

geometriul TvalsaCinoebaze. gairkveva, rogoraa dajgufebuli 

sxvadasxvagvari nakvTebi. _ qvemoTa marcxena kuTxeSi romeli 

jgufia? (da ase Semdeg). 

_ maS, rogori sxvadasxva nakvTebi gvaqvs naswavli? {wertilebi, 

xazebi, mravalkuTxedebi, wreebi, rgolebi da sxvebi}. 

Semdeg Sesruldeba erTi imdagvarive savarjiSo, rogoric 

iyo 95.3-Si. masw.-ma unda miaqcios yuradRReba imas, rom yovel 

moswavles erTxel mainc mouwios gazomvis sakuTari xeliT 

Catarebam. 

:7/5/! `txsbgj! qbtvyfcj! `! Tfebsfcb~/ Sesadareblad zepirad 

eZlevaT jamebi da sxvaobebi (naswavli saxisa). pirvel wreze 

advilebia: 5+7 da 5+6; 4+7 da 3+7; 7 – 1 da 7+4; 5+3 da 

5+5 da misTanani. momdevno wreebze Zneldeba: 6 + 2 da 2 + 9; 

4 + 6 da 7 + 4; 8 _ 3 da 9 – 3; 8 – 4 da 

8 – 2; 3 + 9 da 9 – 3 da misTanani. 

cxadia, masw.-i dasabuTebas ar moiTxovs. moswavles SeuZlia

- 260 - 

iangariSos da ise Seadaros, magram masw.-i klass gaafrTxilebs, 

rom angariSiT ufro meti dro dasWirdebaT da SeiZleba xuT 

TvlaSi ver moaswron. 

g 97. {fqjsj!hbnplmfcb/!vgsptj!!vndsptj/!besf!!hwjbo/! 

:8/1/ 96.7-s yuradRebiT dakvirveba sWirdeba _ TavsatexiviTaa. 

bolos: _ maS, sul ramdeni mravalkuTxedi yofila aq 

daxazuli? daTvaleT! {Tormeti} wertili ramdenia? {erTi} 

xazi ramdenia? {sami}. 

:8/2/!{fqjsj!hbnplmfcb;!23`3-!23`4-!///!-!23`21/! yvelaferi 

gakeTdeba iseve, rogorc 94.1-Si, oRond TormetisaTvis. 

------------------------------------------------------------------------------------ 

II.11. psj! qbsbmfmvsj! )bsb! Ubonjnefwsvmj"*! Tflsfcb! bo! 

hbnplmfcb! tbzpgbdypwsfcp! bnpdbofcTj/! ! obtxbwmj! espjUj! 

njnbsUfcboj! hpofcbTj! xbsnpehfojU! bo! trfnjU! eb! nbUj! 

eblbwTjsfcb!sjhpcjU!sjdywfcUbo/! 

tywbebtywbnpobdfnfcjboj!dysjmjt!bnpljUywb!eb!njtj!Tfwtfcb/ 

aqamde gvxvdeboda mxolod samgvari cxrilebi: nakvTebis formaferisa, 

mravalkuTxedebis gverdebisa da wveroebis raodenobisa da 

Sekreba-gamoklebisa. axla Semogvaqvs zogadad, sxvadasxvamonacemebiani 

cxrili (vgulisxmobT, rom am droisaTvis klasma ukve sakmaod 

icis kiTxva). 

Sekrebisa da gamoklebis sxvadasxva cxrilebis Sevseba ki 

saswavlo wlis bolomde grZeldeba, oRond ZiriTadad mxolod 

saSinao davalebebSi. 

vbewjmftj! psnprnfefcjboj! tbzpgbdypwsfcp! bnpdbofcj/ 

Tanmimdevrul Sekreba-gamoklebas sWirdeba pirveli Sedegis 

damaxsovreba da amitom ufro Znelia, viwyebT jer paraleluri 

Sekreba-gamoklebiT. am SemTxvevaSi ori moqmedeba erTmaneTisagan 

damoukideb-lad sruldeba da amdenad, ufro advilia. 

amocanebs, rogorc yovelTvis, ara mxolod vxsniT, aramed 

winaswar kargad vamuSavebT da pirobas viazrebT, amoxsnis Semdeg 

ki pasuxsac viazrebT. am safexurze es xdeba sagangebo TvalsaCino 

sqemebis meSveobiT. sayofacxovrebo amocanebis Temis erTerTi 

mizani isicaa, rom moswavleebma iswavlon sqemebis Sevseba, Sedgena,

- 261 - 

maTgan monacemTa amokiTxva. amitom amocanis amoxsnisas amocanis 

piroba jer sqemaSi Caiwereba, Semdeg ki moswavle am Sevsebuli 

sqemidan aRadgens amocanis srul pirobas. 

------------------------------------------------------------------------------------- 

:8/3/! hpofcbTj! xbsnpehfojU! bo! trfnjU;! vgsptj! ! vndsptj/! 

besf! ! hwjbo/! hvTjoxjo.hvTjo.eSft.ywbm.{fh.nb{fh/! _ 

warmoidgineT, rom vaJa umcrosia, vidre nana. romelia ufro 

adre dabadebuli? 

Tuki gauWirdebaT, sqema daixatos dafaze: patara biWuna da 

gverdiT didi gogona. 

_ ilia ufrosia, vidre marika. romelia ufro gvian 

dabadebuli? 

_ mamida umcrosia, vidre biZa. romelia ufro gvian 

dabadebuli? 

_ Cveni skolis ezoSi jer naZvi darges da Semdeg gaakeTes 

Robe Tu jer Robe gaakeTes da naZvi Semdeg darges? guSin 

jer maTematika gqondaT da Semdeg musika, Tu jer musika da 

Semdeg maTematika? (da ase Semdeg). 

_ ra hqvia im dRes, romelic iyo dRevandeli dRis win? 

{guSin} imis win? {guSinwin} dRevandeli dRis Semdeg rom iqneba? 

{xval} imis Semdeg? {zeg} kidev Semdeg? {mazeg} romeli 

dRea zegis Semdeg? {mazeg} romeli dRea guSinis Semdeg? {dRes} 

guSinwinis Semdeg? {guSin} mazegis win? {zeg} (da misTanani). 

Tuki gauWirdebaT, sqema daixatos dafaze: erT striqonSi 

kvadratul CarCoebSi Caiwereba: w (guSinwin); g; d; x; z; m. 

:8/4/!psj!yb{jt!hbeblwfUjt!xfsujmfcj/!hbebnlwfUj!npoblwf. 

Ufcj/ dafaze daxazulia (umjobesia, sxvadasxva ferebiT) 

mrudi da monakveTi, romlebic erTmaneTs oTx wertilSi kveTs. 

_ radara xazebia dafaze daxazuli? {monakveTi da mrudi} 

gadakveTs Tu ara isini erTmaneTs? {ki} ramden wertilSi 

gadakveTs? {oTx wertilSi} gamodi da moniSne yvela! erTi, 

ori, sami, oTxi wertili. axla gamovides Tamriko da daxazos 

monakveTi da mrudi ise, rom maTi gadakveTis wertilTa 

raodenoba iyos 6. zurikom ki daxazos monakveTi da mrudi

- 262 - 

ise, rom erTmaneTs ar gadakveTdes! xolo jarjim daxazos ori 

gadamkveTi monakveTi. 

Semdeg msr-Si (97.3): _ radara xazebia daxazuli? {Savi 

texili da wiTeli monakveTi} Tu gadaikveTeba? {ki} ramden 

wertilSi? {sam wertilSi} moniSneT lurjad samive gadakve- 

Tis wertili! axla gverdiT daxazeT ori gadamkveTi monakveTi. 

kidev daxazeT mrudi ise, rom es monakveTebi ar gadakveTos! 

:8/5/!`txsbgj!qbtvyfcj~!`!ÙjUfcjt!Dwfofcb~/ ix. 58.4, oRond 

raodenoba aRwevs Tormetamde. garda amisa, cvlilebaa isic, 

rom aTze meti raodenobis erTbaSad Cveneba SeuZlebelia. 

amitom moswavle aCvenebs or wyebad, magaliTad, Tormetze 

aCvenebs jer aT TiTs da Semdeg _ ors. 

g 98. bUfvmjt!hbnpzpgb/ kvgUfcjU!ebUwmb/!besf!\ hwjbo/! 

:9/1. 97.9-s yuradRebiT dakvirveba sWirdeba _ 

TavsatexiviTaa. bolos: _ maS, sul ramdeni mravalkuTxedi 

yofila aq daxazuli? daTvaleT! {aTi} xazi ramdenia? {oTxi}. 

:9/2/! dbnfujbo! hspwbTj! bUfvmjt! hbnpzpgb/ kvgUfcjU! 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!tbnj~/!nplmfe;!`bUfvmj! 

eb!tbnj~-!bov;!`bUj!eb!tbnj~/!`sbnefojU!nfujb!bU{f@~!`!`tbnjU! 

nfuj~/! hbb{sfcb! tjuzwjtb! Ù.tbn.nfuj~/! ! ! dbnfujt! Tfebsfcb! 

obtxbwm!sjdywfcUbo/ 


cametisaTvis. 

:9/3/! hpofcbTj! xbsnpehfojU! bo! trfnjU;! besf \! hwjbo/! kfs! \ 

Tfnefh/ daismis imdagvarive SekiTxvebi, rogorc 97.2-Si, 

oRond emateba erTi sakiTxi: `ra moxda amis win?~ anda: `ra 

moxda amis Semdeg?~ magaliTad: _ gaixsene, am diliT Sens 

saxlSi rac moxda. piris dabana sauzmis win iyo Tu sauzmis 

Semdeg? fexsacmlis Cacma iyo ufro win Tu sauzme? binis 

karis gaReba iyo ufro Semdeg Tu piris dabana? yvelaze adre 

ra moxda? {sawolidan adgoma} yvelaze gvian? {saxlidan 

gamosvla} kargi, axla miTxari, ra moxda piris dabanis Semdeg; 

ra moxda fexsacmlis Cacmis Semdeg? (da ase Semdeg).

- 263 - 

_ rac ufro adre moxda, is momxdara jer Tu is moxda 

mere? rac ufro gvian moxda? 

:9/4/!Ubwj!:/4.jt!Tfkbnfcb!eb!hbnfpsfcb/!gakeTdeba TiTo-TiTo 

imdagvarive savarjiSo, rogorebic iyo 90.3-Si, 91.3-Si, 93.3-Si 

da 95.3-Si. Sedegebi isev dafaze daiwereba. masw.-ma unda 

miaqcios yuradRReba imas, rom yovel moswavles erTxel mainc 

mouwios gazomvis sakuTari xeliT Catarebam. 

TfojTwob. dajgufebis sakiTxebs gameoreba aRar sWirdeba _ 

drodadro TavisTavad gameordeba saSinao davalebebSi. 

:9/5/!`txsbgj!qbtvyfcj~!`!bohbsjTj/ SekiTxvebia Cveulebrivi 

saangriSo: sams mivumatoT rva; aTs gamovakloT oTxi da ase 

Semdeg _ Tormetis farglebSi. 

:9/7/ masw.-i damatebiT ganmartavs, rom SeiZleba xeliT 

daxatva. Tan Seaxsenebs, rom unda daxaton patara nakvTebi 

(rogorebic iyo, magaliTad, 97.9-Si). 

g 99. {fqjsj!Tflsfcb/!xjo.Tfnefh/!hb{pnwb!DyjsfcjU/ 

::/2/! {fqjsj! Tflsfcb;! 214-! 421-! :5-! 5:-! ///! -! 78-! 87/! ! yvelaferi 

gakeTdeba iseve, rogorc 93.1-Si, oRond cametisaTvis. 

::/3/! hpofcbTj! xbsnpehfojU! bo! trfnjU;! xjo \ Tfnefh/! `sb! 

npyeb!bnjt!xjo!!Tfnefh@~!!!TbsTboxjo.TbsTbo.xfmt.hbjtbe/ 

jer gavimeorebT imdagvarsave SekiTxvebs, rogorebic iyo 

97.2-Si guSinwin-guSin-dRes-xval-zeg-mazeg dReebis Sesaxeb _ 

romeli romlis Semdegaa Tu win. 

_ ra hqvia im weliwads, romelic wlevandeli wlis win 

iyo? {SarSan} imis win? {SarSanwin} romelic wlevandeli 

wlis Semdeg iqneba? {gaisi} romeli welia SarSanwindelis 

Semdeg? {SarSandeli} gaisis win? {wlevandeli} SarSandelis 

Semdeg? {wlevandeli}. 

------------------------------------------------------------------------------------- 

JJJ.21/!hb{pnwb!DyjsfcjU-!DyjsfcjUb!eb!UjUfcjU-!nulbwfmjUb! 

eb! UjUfcjU-! obcjkjUb! eb! ufsgjU/! zwfmb! bn! tb{pnjt! 

dwbmfcbepcb! eb! bsbtj{vtuf/! ! {pnwb! ejej! tbyb{bwjU! `! 

efdjnfusjUb! eb! tboujnfusjU/! ejej! tbyb{bwjt! ebobzpgfcjt!

- 264 - 

xbljUywb/! ypmp! hb{pnwb! Upljt! tbTvbmfcjU-! nbU! Tpsjt! nsvef! 

yb{jtb-!vlwf!wbtxbwmfU/! 

gazomvis TemaSic mivyvebiT bunebriv istoriul gzas. moswavle 

TviTon unda darwmundes, rom saWiroa myari da ucvleli sazomi 

erTeulebi, rom saxazaviT zomva mosaxerxebelia. Tumca, yofacxovrebaSic 

Cven xSirad gvWirdeba xolme Zveli sazomebis 

gamoyeneba (mtkavelis, Tokisa da sxva). amitom amis codna 

TavisTavadac saWiroa. 

sigrZis or-ori erTeuli saWiroa imisaTvis, raTa gavzomoT 

pirveli, didi erTeuliT gazomvis Semdeg morCenili naSTi. amis 

saWiroebasac moswavle Tavad Seejaxeba, roca ver ityvis: sigrZe 5 

Cxiria Tu 6. 

yvela safexurze gazomvis Sedegebs dafaze vwerT. 

------------------------------------------------------------------------------------ 

::/4/! sbjnft! hb{pnwb! DyjsfcjU;! njtj! DyjsfcjU! Tfwtfcjt! 

tbTvbmfcjU! )vobTUpe*/ masw.-s xelSi Cxirebis kona ukavia 

(TxuTmetiode Cxiri). _ modi axla am fanjris rafis sigrZe 

am CxirebiT gavzomoT! vin gazomavs? (moswavle daawyobs 

Cxirebs erT swor xazze rafis gaswvriv, masw.-i yuradRebas 

gaamaxvilebs imaze, rom Cxirebi sworad unda iyos dawyobili, 

Tan erTmaneTze mijriT). ramdeni Cxiri yofila rafis sigrZe? 

{rafis sigrZea Tormeti Cxiri}. 

aseve gaizomeba kidev ori ram. 

am gakveTilze naSTi ar unda morCes. amitom masw.-s winaswar 

ise unda hqondes SerCeuli gasazomi nivTebi, rom maTi sigrZe 

iyos Cxirebis mTeli raodenobis toli, Tanac araumetes 13isa! 

garda amisa, gasazomi nivTi iseTi unda iyos, rom masze 

Cxirebi kargad lagdebodes. 

::/5/! `txsbgj! qbtvyfcj! `! lblvo.ubTfcj~/ ix. 18.4, oRond 

raodenoba aRwevs 13-mde. 

g 100. {fqjsj!hbnplmfcb/!xjo!!Tfnefh/!hb{pnwb!DyjsfcjU/!!! 

211/2/!{fqjsj!hbnplmfcb;!24`4-!24`5-!///!-!24`21/!!!!!yvelaferi 

gakeTdeba iseve, rogorc 94.1-Si, oRond cametisaTvis.

- 265 - 

211/3/!hpofcbTj!xbsnpehfojU!bo!trfnjU;!xjo!!Tfnefh/!`ft!sjt! 

Tfnefh!!xjo!npyeb@~! jer gavimeorebT imdagvarsave SekiTxvebs, 

rogorebic iyo 97.2-Si guSinwin-guSin-dRes-xval-zeg-mazeg 

dReebis Sesaxeb _ romeli romlis Semdegaa Tu win. Semdeg 

daemateba Seb-runebuli SekiTxvebic: _ guSin romeli dRis 

winaa? romeli dRis Semdegaa? zeg romeli dRis winaa? romeli 

dRis Semdegaa? (Tuki gauWirdebaT, dafaze daixazeba iseTive 

sqema, rogoric 97.2-Si). 

Semdeg gakeTdeba yvelaferi aseve, oRond weliwadis 

droTaTvis. jer SekiTxvebia: _ weliwadis romeli droa 

gazafxulis Semdeg? gazafxulis win? Semdeg ki _ 

Sebrunebulebi: _ Semodgoma ris Semdegaa? ris winaa? 

albaT gauWirdebaT, amitom dafaze daixazeba wriuli sqema 

da zed daewereba: g, z, S, T (T iyos `TeTri zamTari, 

Tovli~...). 

211/4/! sbjnft! hb{pnwb! DyjsfcjU;! njtj! DyjsfcjU! Tfwtfcjt! 

tbTvbmfcjU! eb! obTUjt! hb{pnwb! UjUfcjU/! ! ! pirveli savarjiSo 

keTdeba iseve, rogorc 99.3-Si. meore ki ise SeirCeva, rom 

morCes naSTi. _ maS ramdeni Cxiris sigrZea dafa? rvisa Tu 

cxrisa? {arc rvaa swori da arc cxra _ rvaze metia, cxraze 

naklebia} aba ra vqnaT, rogor gavzomoT dafa? 

gamoiTqmeba sxvadasxva azri. masw.-i miaxvedrebs, rom 

SeiZleba TiTebis moSvelieba. magram rogor? morCenili naSTi 

gaizomos TiTebiT (erTmaneTze mityupebuli TiTebis siganiT). 

miviRebT, rom dafis sigrZea rva Cxiri da sami TiTi. 

aseve gaizomeba kidev erTi ram. 

211/5/!`txsbgj!qbtvyfcj!`!ÙjUfcjt!Dwfofcb~/ ix. 97.4, oRond 

raodenoba aRwevs cametamde. 

211/9. marcxena svetSi oro nakvTi erTmaneTs edeba _ 

TiTqos, ìkribeba~; marjvenaSi ki pirvels àkldeba~ meore. 

g 101. bUfvm/!hbnpz/!Ù.pUy.nfuj~/!Uvlj!i=a, a>o, maSin i>o. 

212/1/ 100.7 _ SemoqmedebiTia. 

212/2/! UpUynfujbo! hspwbTj! bUfvmjt! hbnpzpgb/! kvgUfcjU!

- 266 - 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!pUyj~/!nplmfe;!`bUfvmj! 

eb! pUyj~-! bov;! `bUj! eb! pUyj~/! `sbnefojU! nfujb! bU{f@~! `! 

`pUyjU! nfuj~/! hbb{sfcb! tjuzwjtb! Ù.pUy.nfuj~/! UpUynfujt! 

Tfebsfcb! obtxbwm! sjdywfcUbo/! yvelaferi gakeTdeba iseve, 

rogorc 92.1-Si, oRond ToTxmetisaTvis. 

212/3/! obtxbwm! espjU! njnbsUfcbUb! eblbwTjsfcb! sjhpcjU! 

sjdywfcUbo/! ganvixilavT adre ukve garCeul suraTs 101.2. 

moswavleebma am suraTs zedve unda daaweron cifrebi _ im 

TanmimdevrobiT, rogorc yvelaferi moxda. ese igi, cifri 0 

unda daaweron gogonas saxlis Sesasvlels. _ nuli imitom, 

rom saxlTan jer ar dawyebula siaruli. yvelaze adre rasTan 

gaivlis gogona amis Semdeg? {RobesTan} ra cifri unda 

davweroT RobesTan? {erTiani} meored rasTan gaivlis gogona? 

{cxvrebTan} daawereT iq Sesabamisi cifri! mesamed rasTan 

gaivlis gogona? {ZroxasTan rom mosaxvevia} daawereT iq 

Sesabamisi cifri! (da ase Semdeg). ukanasknelad, yvelaze gvian 

rasTan iqneba gogona? 

Semdeg daxuron msr da Seecadon, warmodgeniT upasuxon 

imave SekiTxvebs, romlebic mocemulia 86.2-Si. 

212/4/!tjhs[ffc{f;!!Uvlj!!j = b!!eb!!b > p-!!nbTjo!!j > p/! 

arsebiTad, klass es sakiTxi ukve ecodineba. es 

sagakveTilo monakveTi ZiriTadad naswavlis gameorebasa da 

ganmtkicebas eTmoba. gakeTdeba ori imdagvari savarjiSo, 

rogorebic iyo 95.3-Si, oRond, araarsebiTi cvlilebiT (95.3isa 

da 101.3-is saTaurebis Sesaba-misad). 

Semdeg gakeTdeba ori imdagvarive savarjiSo, rogorebic 

iyo 100.3-Si, oRond erTi siaxliT: masw.-i gazomvis Sedegs 

dafaze dawers. 

Canaweris nimuSi: karadis sigane _ 8 Cxiri da 8 TiTi. 

TfojTwob/ masw.-ma gasazomi nivTebi ise unda SearCios, rom 

9-ze meti Cxiri ar dasWirdes, radganac ufro didi ricxvebis 

Cawera jer ar gviswavlebia. 

212/5/!`txsbgj!qbtvyfcj!`!xjo!Uv!Tfnefh~/ SekiTxvebi Seexeba

- 267 - 

droiT mimarTebebs: _ romeli dRea guSinis Semdeg? romeli 

dRis winaa zeg? weliwadis romeli droa zamTris win? ris 

Semdegaa zafxuli? (da misTanani). 

g 102. {fqjsj!Tflsfcb/!Tfwt/!dysjmj/!hb{pnwb!DyjsfcjU/! 

213/1/ 101.7-Si saWiroa kanonzomierebis sityvieri Camoyalibebac: 

6 Sekruli texili da 5 gaxsnili texili; 6 Sekruli 

mrudi da 4 gaxsnili mrudi; 3 xuTkuTxedi da 6 eqvskuTxedi; 

3 Sekruli xazi da 8 gaxsnili xazi. 

212/9/ dafaze unda daixazos da gairCes. 

212/:. marcxena svetSi oro nakvTi erTmaneTs edeba _ 

TiTqos, ìkribeba~; marjvenaSi ki pirvels àkldeba~ meore. 

213/2/! {fqjsj! Tflsfcb;! 215-! 521-! :6-! 6:-! 97-! 79-! 88/ yvelaferi 

gakeTdeba iseve, rogorc 93.1-Si, oRond ToTxmetisaTvis. 

213/3/!tywbebtywbnpobdfnfcjboj!Tfwtfcvmj!dysjmjt!tusjrpoj! 

eb!twfuj/!_ gaixseneT, rac viswavleT Cven _ ra SeiZleba iyos 

cxrilis ujredebSi? {naxatebi, nakvTebi, ricxvebi...} cxrilis 

ujredebSi sityvebic SeiZleba eweros. kanonzomiereba ki 

igivea, rac viciT _ striqonebisa da svetebis mixedviT. ai 

vnaxoT gakveTilebis cxrili (msr-s 102.2). Tqveni cxrili 

sxvanairia, es sxva pirvelklaselebis cxrilia. ramdeni sveti 

aqvs am cxrils? {eqvsi} striqoni? {imdenive, eqvsi} daakvirdiT 

jer pirvel striqons. Sig ra weria? (waikiTxon kviris 

dReebi). risTvisaa aq kviris dReebis saxelebi? {romel dRes 

romeli gakveTilebia} naxeT axla pirveli sveti. ra ricxvebi 

weria Sig? isini gviCvenebs, meramdene gakveTilia: pirveli, 

meore, mesame, meoTxe, mexuTe. erT dReSi xuT gakveTilze meti 

pirvelklaselebs Tu aqvT? {ara} amitom araa saWiro ufro 

meti striqoni. mexuTeklaselebis gakveTilebis cxrili rom 

yofiliyo, ramdeni striqoni dasWir-deboda? {Svidi an rva, 

imitom, rom mexuTeklaselebs erT dReSi eqvsi da Svidi 

gakveTili aqvT} kargi, axla naxeT meore sveti. risi svetia 

igi? {orSabaTisa} am svetSi ra weria? {orSabaTs romeli 

gakveTilebia} ... (da ase Semdeg _ meeqvse svetamde).

- 268 - 

_ axla vnaxoT striqonebi. meore striqoni risi 

striqonia? {pirveli gakveTilebisa} diax, masSi weria, 

TviTeul dRes romelia pirveli gakveTili. mesame striqoni 

risi striqonia? ... (da ase Semdeg _ meeqvse striqonamde). 

_ ramdeni dRea kviraSi? CamoTvaleT yvela! ratom ara aqvs 

cxrils SabaTisa da kviris svetebi? {imitom, rom SabaTs da 

kviras ar swavloben, klass gakveTilebi ar utardeba} romel 

ujredebSia gasmuli xazi? {qveda striqonis mesame, mexuTe da 

meeqvse ujredebSi} am sam ujredSi ratomaa gasmuli xazi? {im 

dReebSi mexuTe gakveTili ara aqvT, oTxi gakveTili aqvT} 

romeli dReebSi aqvT xuTi gakveTili? oTxi gakveTili? sami 

an ori gakveTili Tu aqvT romelime dRes? 

213/4/! sbjnft! hb{pnwb! DyjsfcjU;! fsUj! Dyjsjt! hbebubojt! 

tbTvbmfcjU! eb! obTUjt! hb{pnwb! UjUfcjU/! _ gvinda, CxirebiT 

gavzomoT am magidis simaRle. rogor gavzomoT? {Cxirebi 

davawyoT} ki magram, rom Camocvivdeba? {saTiTaod davikaoT 

bavSvebma} Zalian cudia ase gazomva, aTma bavSvma erTad unda 

imuSaos, TiTqos didi rame iyos! sinamdvileSi ki sul 

advilia! ra didi rame unda?! erTi bavSvic ki advilad 

gazomavs! vin mixvdeba, rogor? 

gamoiTqmeba sxvadasxva azrebi. masw.-i miaxvedrebs: erTi 

Cxiris gadatana SeiZleba. erTi Cxiric ki sakmarisia! 

ase gavazomvinebT ramdenime sxvadasxva rames. oRond meore 

jerobaze Cxiris nacvlad kalmistars gamovayenebinebT, mesameze 

_ fanqars. naSTs kvlav TiTebiT gazomaven. gazomvis Sedegebi 

dafaze daiwereba (amitom sigrZe ar unda iyos 9-ze meti). 

213/5/! `txsbgj! qbtvyfcj! `! sjdywjt! eblblvofcb~/ am TamaSs 

aRvwerT sruli saxiT, anu ocdacxris farglebSi. cxadia, 

yovel gaTamaSebaze masw.-i im farglebiT SemoizRudeba, rac 

klass im droisaTvis naswavli eqneba. 

dafaze daxazulia ori Setyupebuli CarCo (ujredebi). 

erTSi weria: àTeulebi~, meoreSi weria: `naSTi~. masw.-i 

maCvenebeli joxis wveriT akakunebs am ujredebSi, moswavlem 

ki unda gamoicnos Sesabamisi ricxvi. magaliTad, masw.-ma

- 269 - 

daakakuna erTxel àTeulebSi~ da samjer _ `naSTSi~. 

moswavlem unda Tqvas: cameti. sxva magaliTebi: orjer 

àTeulebSi~ da xuTjer _ `naSTSi~ {ocdaxuTi}, arcerTxel 

àTeulebSi~ da rvajer _ `naSTSi~ {rva}, orjer àTeulebSi~ 

da arcerTxel _ `naSTSi~ {oci}, erTxel àTeulebSi~ da 

Svidjer _ `naSTSi~ {Cvidmeti}, erTxel àTeulebSi~ da 

arcerTxel _ `naSTSi~ {aTi} da ase Semdeg. 

amjerad, cxadia, SemovifarglebiT 14-iT. 

213/7/ saWiroa axsna, Tu sad ra unda Caiweros: zeda striqonSi 

iwereba magidis sigane, qvedaSi _ skamis simaRle. ramdeni 

fanqaric iqneba sigrZe _ is ricxvi unda Caiweros sityvis 

`fanqari~ win cariel adgilas; ramdeni TiTic iqneba naSTi _ 

is ricxvi unda Caiweros sityvis `TiTi~ win cariel adgilas. 

dafaze unda daiweros nimuSi da ise aixsnas. 

213/9/ yvela nakvTi mravalkuTxedia, garda erTisa (wris 

CamonaWrisa). swored igi arRvevs kanonzomierebas. 

g 103. {fqjsj!hbnplmfcb/!{pnwb!nulbwfmjUb!eb!UjUjU/ 

214/2/! {fqjsj! hbnplmfcb;! 25`5-! 25`6-! ///! -! 25`21/! yvelaferi 

gakeTdeba iseve, rogorc 94.1-Si, oRond ToTxmetisaTvis. 

214/3/!tywbebtywbnpobdfnfcjboj!dysjmjebo!npobdfnUb!bnpljU.! 

ywb/! ganvixilavT kvlav gakveTilebis cxrils 102.2. 

sxvadasxvagvar SekiTxvaTa nimuSebi: 

1) romelia oTxSabaTs meore gakveTili? {qarTuli} saidan 

ici? {oTxSabaTis svetisa da meore gakveTilebis striqonis 

gadakveTis ujredSi weria}. 

2) paraskevs qarTulebi meramdene gakveTilebia? {pirveli da 

meore}. 

3) romel dResaa musika meoTxe gakveTili? {xuTSabaTs}. 

4) romel dReebSi aqvT xatva? romel dReebSi ara aqvT 

fizkultura? 

5) romelia is sagnebi, romlebic klass yoveldRiurad aqvs? 

6) sul ramden sxvadasxva sagans swavloben bavSvebi? {xuTs} 

romlebs?

- 270 - 

7) erT kviraSi ramdenjer utardebaT maTematika? musika? 

xatva? qarTuli? fizkultura? daTvaleT! 

214/4/! {pnwb! nulbwfmjUb! eb! UjUjU/! _ vTqvaT, varT sadme 

mindorSi an zRvis napiras. Tan ar gvaqvs arc fanqari, arc 

Cxiri da arc raime amgvari nivTi. rogor gavzomoT rame? ra 

gvaqvs yovelTvis Tan? {Cveni sakuTari xelebi} xelebiT 

SegviZlia gavzomoT. Cxirisa da fanqris nacvlad gavzomoT 

mtkaveliT. ai, mtkaveli esaa! ai ceris am wveridan nekis am 

wveramdea mtkaveli. aba yvelam gaSaleT Tqveni mtkaveli! 

TiTebi kargad unda iyos gaWimuli, ise araa mtkaveli, ai es 

mtkaveli araa! (masw.-i cotaTi moxris TiTebs). axla gamovides 

miSiko da mtkaveliT gazomos am magidis sigane (naSTi kvlav 

TiTebiT gaizomeba). 

gazomvis Sedegi dafaze daiwereba. 

aseve Sesruldeba kidev sami gazomva. 

214/5/! `txsbgj! qbtvyfcj~! `! bohbsjTj/ SekiTxvebia 

Cveulebrivi saangriSo: sams mivumatoT rva; aTs gamovakloT 

oTxi da ase Semdeg _ ToTxmetis farglebSi. 

g 104. bUfvmjt!hbnpzpgb/!Ù.yvU.nfuj~/!{pnwb!obcjkjU/! 

215/2/! UyvUnfujbo! hspwbTj! bUfvmjt! hbnpzpgb/! kvgUfcjU! 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!yvUj~/!nplmfe;!`bUfvmj! 

eb! yvUj~-! bov;! `bUj! eb! yvUj~/! `sbnefojU! nfujb! bU{f@~! `! 

`yvUjU! nfuj~/! hbb{sfcb! tjuzwjtb! Ù.yvU.nfuj~/! UyvUnfujt! 


rogorc 92.1-Si, oRond TxuTmetisaTvis. 

215/3/!tywbebtywbnpobdfnfcjboj!dysjmjt!Tfwtfcb/!!_ vnaxoT es 

cxrili (104.2). ramdeni sveti aqvs mas? ramdeni striqoni? 

misi svetebi araa Sevsebuli, axla unda SevavsoT. vin mixvdeba, 

risi striqonia meore striqoni? {moswavleebis wlovanebaTa} 

am moswavleebis saxelebi sad unda eweros? {pirvel 

striqonSi, yoveli svetis zeda ujredSi}. risia meore 

striqoni? {moswavleTa simaRleTa} mesame striqoni? 

{moswavleTa mxrebis siganeTa}. ramdeni moswavle gvWirdeba am

- 271 - 

cxrilis Sesavsebad? {xuTi} ratom? {xuTi carieli svetia}. 

saTiTaod gamoviyvanT xuT moswavles, romlebic wina 

gakveTilebze TamaSebSi imarjvebdnen. svetis zeda ujredSi 

klasi Cawers moswavlis saxelis pirvel asos, meore ujredSi 

_ wlovanebas ase: 6 (igulisxmeba weli). Semdeg masw.-i 

sakuTari mtkaveliTa da TiTebiT gazomavs moswavlis simaRles 

da Sedegi Caiwereba Sesabamis ujredSi ase: 7 m 5 T. aseve 

gaizomeba da Caiwereba mxrebis siganec. 

215/4/! {pnwb! obcjkjUb! eb! ufsgjU/! Tuki kargi daria, es 

sagakveTilo monakveTi SeiZleba ezoSi Catardes. 

_ warmovidginoT, rom ezoSi varT da gvWirdeba saTamaSo 

moednis sigrZisa da siganis gazomva. mtkaveliT daviwyoT 

gazomva? {ara, Zalian mouxerxebeli iqneba!} maS riTi jobia 

gavzomoT? {nabijebiT} naSTi ki jobia gavzomoT terfebiT. ai, 

esaa terfi! 

nabijiTa da terfiT gaizomeba saklaso oTaxis sigrZe da 

sigane an ezoSi raime manZilebi. 

215/5/! `txsbgj! qbtvyfcj! `! dysjmjebo~/ am TamaSisaTvis 

umjobesia, klasSi ekidos erTi didi cxrili gakveTilebisa 

(SeiZleba sxvac, sityvebiani). Tuki ara gvaqvs, yvelam 

gadaSalos msr 102.2 cxrilze. SekiTxvebis nimuSebi 

mocemulia 103.2-Si. 

g 105. {fqjsj!Tflsfcb/!{pnwb!ejej!tbyb{bwjU!`!en.jUb!eb!tn.jU/! 

216/1. 104.8-Si kanonzomierebaa: yvela sxvaobaSi saklebia 

erTidaigive 9, maklebebi ki aris 9, 8, 7 da ase Semdeg. 

104.9-Si albaT mcdarad eqnebaT Semoxazuli. saWiroa 

gaimarTos msjeloba: ar viciT, romeli iyo ufro grZeli. es 

imitom, rom mamis mtkaveli ufro didia, vidre Svilisa da 

misi 6 mtkaveli SeiZleba ufro didi yofiliyo, vidre Svilis 

9 mtkaveli. am amocanas pasuxi ara aqvs. 

216/2/! {fqjsj! Tflsfcb;! 10+5, 5+10, 9+6, 6+9, 8+7, 7+8/! yvelaferi 

gakeTdeba iseve, rogorc 93.1-Si, oRond TxuTmetisaTvis. 

216/3/!psj!qbsbmfmvsj!Tflsfcb!tbzpgbdypwsfcp!bnpdbofcTj/

- 272 - 

damuSavdeba amocana: erT oTaxSi 5 skami da 2 magida idga, 

meore oTaxSi ki _ 4 skami da 1 magida. ramdeni skami da 

ramdeni magida mdgara orive oTaxSi erTad? 

masw.-i Tan dafaze daxazavs or cariel CarCos (rogorcaa 

msr-Si 105.2, erTi naxevari). _ warmovidginoT, rom es viTom 

erTi oTaxia, es _ meore. ramdeni skamia pirvel oTaxSi? {5} 

CavweroT aq 5 s (Cawers zeda naxevarSi) ramdeni magidaa am 

oTaxSi? {2} CavweroT aq 2 m (Cawers qveda naxevarSi). ras 

aRniSnavs es aso s? {skams} 5-iani? {rom 5 skamia} es aso m? 

{magidas} 2-iani? {rom 2 magidaa}. 

aseve Seivseba meore CarCoc. _ SexedeT am CarCoebs da 

mipasuxeT: ramdeni skamia pirvel oTaxSi? meoreSi? ramdeni magidaa 

pirvel oTaxSi? meoreSi? ras gvekiTxeba amocana? Tamriko, 

ucqire am naxazs da srulad waikiTxe amocanis piroba! 

kargi, axla amovxsnaT amocana. ramdeni skami yofila orive 

oTaxSi? {5 + 4 = 9} davweroT aq 9 s (dawers CarCoebis 

gverdiT, marjvniv, cariel adgilas, moswavleebi msr-Si 

weren). ramdeni magida yofila orive oTaxSi? {2+1=3} 

davweroT aq 3 m. 

_ vin waikiTxavs amocanis pasuxs srulad? 

aseve damuSavdeba kidev erTi amdagvari amocana, oRond 

moswavleebi Tavad Seavseben msr-Si cariel CarCoebs (105.2). 

216/4/! nulbwfmjtb! eb! UjUjt! dwbmfcbepcb! eb! bsbtj{vtuf/! 

{pnwb! ejej! tbyb{bwjU! `! efdjnfusjUb! eb! tboujnfusjU/! ejej! 

tbyb{bwjt! ebobzpgfcjt! xbljUywb-! nbhbmjUbe;!6!efdjnfusj!eb!4! 

tboujnfusj/! ! Tanafardoba 1 dm = 10 sm jer araa saWiro, 

jer-jerobiT dm cvlis mtkavels, xolo sm _ TiTs. 

mtkaveliTa da TiTiT gaizomeba, magaliTad, fanjris rafa, 

oRond erTi siaxliT: gazomaven sxvadasxva moswavleebi _ 

SevarCevT taniT didad gansxvavebul xuT moswavles. Semdeg 

masw.-ic gazomavs. yvela Sedegi dafaze erT svetSi Camoiwereba. 

klasi darwmundeba, rom sul sxvadasxva ricxvebi miiReba. 

_ ratom miiReT es sxvadasxva sigrZeebi? {imitom, rom 

bavSvebsac da masw.-sac sxvadasxva zomis mtkavelebi da TiTebi

- 273 - 

aqvT, zogs _ didi, zogs _ patara} romelia amaTgan rafis 

sigrZe? {ar viciT} aba ra vqnaT, rogor gavigoT sigrZe? 

gavzomoT saxazaviT! ai am saxazavze ricxvebi weria. ai am 

mokle monakveTebs Soris manZils hqvia decimetri. mtkavelis 

nacvlad gavzomoT decimetriT. am patar-patarebs ki 

santimetrebi hqvia. rad gvinda santimetrebi? {naSTis gasazomad, 

TiTebis nacvlad} aba gavzomoT Cveni rafis sigrZe! 

masw.-i aCvenebs, rogor unda saklaso metriani saxazavis 

xmareba gazomvisas. decimetrebi daTvalon, warwerebs jerjerobiT 

nu miaqceven yuradRebas. naSTis santimetrebic 

daTvalon. Sedegi dafaze daiwereba: 

rafis sigrZe = 7 dm 2 sm 

_ ras aRniSnavs es dm? {decimetrs} es sm? {santimetrs} 7iani? 

{rom 7 decimetria} 2-iani? {rom kidev 2 santimetria}. 

sxvadasxva moswavleebma da masw.-ma rom mtkavelebiT gazomes, 

sigrZeebi sxvadasxva miviReT _ ai, aq xom gviweria! decimetrebiT 

rom gaezomaT sxvadasxva moswavleebs, sigrZeebs 

sxvadasxvas miviRebdiT Tu erTnairs? {erTnairs} ratom? 

{decimetri yvela erTnairia, sulerTia, vin zomavs!}. 

216/5/!`txsbgj!qbtvyfcj!`!lblvo.ubTfcj~/ 

ix. 18.4, oRond raodenoba aRwevs 15-mde. 

--------------------------------------------------------------------------------- 

216/9/ daiwyo `logikuri kvadratis~ amocanebis grZeli jaWvedi. 

rogorc yvela sxva jaWvedi, igic TandaTanobiTi garTulebiTaa 

agebuli. ZiriTadi kanonzomiereba erTidaigivea: cxrilSi imdeni 

sxvadasxvagvari sagania (nakvTi an maTi wyvili, naxati, aso, sityva...), 

ramdenicaa sveti (an striqoni). mTavari isaa, rom TiTo svetSica 

da TiTo striqonSic zustad TiTo saxis sagania. 

----------------------------------------------------------------------------------g 

106. {fqjsj!hbnplmfcb/!{pnwb!ejej!tbyb{bwjU/!!! 

217/1/ 216/8 dafaze gairCeva. 

216/9-Si kanonzomierebis sityvierad Camoyalibeba Znelia da 

araa saWiro. magram amocana ki kargad unda gairCes. igi albaT, 

moswavleTa umravlesobas ar eqneba kargad amoxsnili.

- 274 - 

cxrilSi sul samgvari nakvTia: wiTeli samkuTxedi, Savi 

wre da ruxi kvadrati (ferebi moswavles Sin gauadvilebda 

amoxsnas). ujredebSi maTi wyvilebia. kanonzomierebis misaxvedrad 

advilia amgvari midgoma: _ gaadevneT Tvali pirvel svets. 

ra aris erTidaigive? ra icvleba? {pirvel svetSi samkuTxedi 

yvelganaa, misi mewyvile ki icvleba: samkuTxedi, wre, 

kvadrati}. aseve meore svetSi? {yvelganaa wre, misi mewyvile 

ki icvleba: samkuTxedi, wre, kvadrati}. aseve mesame svetSi 

rogor iqneba? {yvelgan unda iyos kvadrati, misi mewyvile ki 

icvleba: igive samkuTxedi, wre, kvadrati}. 

aseTive kanonzomierebaa striqonebSic, oRond igivea ara 

pirveli, aramed meore nakvTi, pirveli ki iseve icvleba: 

samkuTxedi, wre, kvadrati. Tumca striqonebisTvis Tvalis 

gadevneba ufro Znelia, nakleb TvalsaCinod Cans. 

kanonzomierebis zogadad sityvierad gaazreba Znelia, yvelaferi 

mxolod mocemuli konkretuli cxrilis TvalsaCino doneze 

unda gairkves. 

qveda Sua ujredSi unda exazos wre da kvadrati; marjvena 

qveda ujredSi _ kvadrati da kvadrati. 

216/:-Si moswavleebs wesiT aRar unda SeSlodaT ise, rogorc 

SeeSlebodaT 104.9-Si. msjeloba aqac imdagvarive unda Catardes. 

217/2/! {fqjsj! hbnplmfcb;! 26`6-! 26`7-! ///! -! 26`21/! yvelaferi 

gakeTdeba iseve, rogorc 94.1-Si, oRond TxuTmetisaTvis. 

217/3/!psj!qbsbmfmvsj!Tflsfcb!tbzpgbdypwsfcp!bnpdbofcTj-! 

spdb! npobxjmfpct! ovmj! )bsdfsUj*/ iseve, rogorc 105.2-Si, 

Se-kiTxvebiT damuSavdeba da amoixsneba ori amocana. pirveli _ 

da-faze da Tan msr-Si, meore _ mxolod msr-Si (106.2). 

siaxlea sit-yva àrcerTi~. sqemebSi (CarCoSi) mis Sesabamisad 

unda Caiweros 0. 

1) gogonas TaigulSi 8 gvirila da 4 mixakia, xolo misi 

Zmis TaigulSi arcerTi gvirila araa, mixaki ki 5-ia. ramdeni 

gvirila da ramdeni mixakia orives TaigulebSi erTad? 

2) moswavlem daxaza wiTeli da Savi nakvTebi: 6 wiTeli 

kvadrati, 4 wiTeli wre da 3 Savi kvadrati (Savi wreebi ki

- 275 - 

ar dauxazavs). ramdeni kvadrati da ramdeni wre dauxazavs 

sul moswavles? 

217/4/! obcjkjtb! eb! ufsgjt! dwbmfcbepcb! eb! bsbtj{vtuf/! 

{pnwb! lwmbw! ejej! tbyb{bwjU! `! efdjnfusjUb! eb! tboujnfusjU/!!! 

jer gakeTdeba igive, rac 105.3-is dasawyisSi, oRond nabijebiTa 

da terfebiT gaizomeba saklaso oTaxSi or skams Soris 

manZili (romelic ise SeirCeva, rom moswavlis 9 nabijze meti 

ar iyos!). bolos masw.-i ityvis: _ saxazaviT amgvari manZilebis 

gazomva mouxerxebelia. saWiroa xveula an sxva sazomi 

xelsawyo. amas SemdgomSi viswavliT. axla isev decimetrebiTa 

da santimetrebiT gazomvaSi vivarjiSoT. vin gviCvenebs am 

saxazavze, saidan sadamdea decimetri? santimetri? 

Semdeg Sesruldeba ramdenime gazomva decimetrebiT. 

Sedegebi (sigrZeebi ar unda iyos 9-ze meti) dafaze daiwereba, 

iseve rogorc 105.3-Si. 

_ sxva qalaqSi, sxva skolaSi, sxva klasSi sxva saxazavi 

aqvT. Cveni saxazavis decimetri rom SevadaroT sxva saxazavis 

decimetrs _ rogor ggoniaT, romeli iqneba meti? {toli 

iqneba, yvela saxazavis decimetrebi tolia} saxazavs rom ufro 

mokle an ufro grZeli decimetri hqondes, is saxazavi gafuWebuli 

iqneboda! yvela karg saxazavze decimetrebi tolia. 

Tanac, sulerTia, vin zomavs, yvela decimetri zustad tolia! 

217/5/!`txsbgj!qbtvyfcj!`!UjUfcjt!Dwfofcb~/ ix. 97.4, oRond 

raodenoba aRwevs TxuTmetamde. 

217/7/! kanonzomierebaa: oTxkuTxedi, xuTkuTxedi, eqvskuTxedi, 

SvidkuTxedi... matulobs zeda qimebis raodenoba. 

217/9 imdagvaria, rogoric iyo 105.8, oRond samis nacvlad 

oTxia, nakvTebica da svet-striqonebic. cariel ujredebSi 

unda iyos wyvilebi jvari-SvidkuTxedi (qvemoT); SvidkuTxedijvari 

da oTxkuTxedi-jvari (meoTxe svetSi). 

g 107. bUfvmjt!hbnpzpgb/ Ù.frwt.nfuj~/!yb{fcjt!ebkhvg/! 

218/2/! Ufrwtnfujbo! hspwbTj! bUfvmjt! hbnpzpgb/ kvgUfcjU! 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!frwtj~/!nplmfe;!`bUfvmj!

- 276 - 

eb! frwtj~-! bov;! `bUj! eb! frwtj~/! `sbnefojU! nfujb! bU{f@~! `! 

`frwtjU! nfuj~/! hbb{sfcb! tjuzwjtb! Ù.frwt.nfuj~/! Ufrwtnfujt! 


rogorc 92.1-Si, oRond TeqvsmetisaTvis. 

218/3/ psj! qbsbmfmvsj! hbnplmfcb! tbzpgbdypwsfcp! bnpdb. 

ofcTj/! iseve, rogorc 105.2-Si, SekiTxvebiT damuSavdeba da 

amoixsneba ori amocana. pirveli _ dafaze da Tan msr-Si, 

meore _ mxolod msr-Si (107.2). oRond Sekrebis nacvlad 

gamoklebaa. amitom sqemaSi carieli CarCo mxolod erTia, masSi 

Caiwereba is monacemebi, rac Tavidan iyo (ramdeni Citic ijda 

totze Tavidan da sxva). CarCos marjvniv cariel adgilas 

gamoklebis niSniT miewereba is, Tu ramden-ramdeni daaklda. 

Semdeg Camoismeva Sveuli monakveTi da kidev ufro marjvniv 

miewereba pasuxebi. magaliTad: 

1) gakveTilze 8 gogona da 7 biWuna iyo. zaris darekvis 

Semdeg gareT gavida 4 gogona da 6 biWuna. ramdeni gogona da 

ramdeni biWuna darCenila saklaso oTaxSi? 

damuSaveba iseTivea, rogoric 105.2-Si, Canaweri ki aseTia: 

8 g – 4 g 4 g 

7 b – 6 b 1 b 

Sorena, ucqire am naxazs da srulad waikiTxe amocanis 

piroba! kargi, axla amovxsnaT amocana! 

2) moswavles unda daexaza 6 monakveTi da 4 wre. man 

daxaza 6 monakveTi da 2 wre. ramdeni monakveTi da ramdeni 

wre darCenia mas dasaxazad? (am meore amocanaSi mcire 

siaxlea: erTerTi pasuxia 0). 

218/4/! UwbmtbDjope! npdfnvmj! yb{fcjt! ebkhvgfcb;! npoblwf. 

Ufcj-!ufyjmfcj! eb!tywfcj!`! nsvefcj/! es sakiTxi klass ukve 

ecodineba. siaxlea TviTeuli xazis daTvla da Sedegebis 

Cawera siaSi. 

SevasrulebT erT gazomvas decimetrebiTa da santimetrebiT 

(iseve, rogorc 105.3-Si). Sedegi dafaze aRar davweroT. amitom 

decimetrebis raodenoba iyos 14, 15 an 16. ese igi, saWiro 

iqneba metriani saxazavis gadacoceba! esec erTi mcire siaxlea.

- 277 - 

Semdeg Sesruldeba msr-s 107.3. yuradRebiT daTvlian: 

yvela monakveTs, yvela texils da yvela mruds. daTvlisas 

unda gamoiyenon feradi fanqrebi, uamisod Zneli iqneba. monakveTebis 

daTvlisas TviTeuli daTvlili monakveTi moniSnon, 

magaliTad, mwvane wertiliT; texilebi _ wiTeli da mrudebi 

_ lurji wertilebiT. 

raki moswavleebma orniSna ricxvebis wera jer ar ician, 

daTvlis Sedegebi Caweron `dominoseburi~ wertilebiT. 

218/5/! `txsbgj! qbtvyfcj~! `! bohbsjTj/ SekiTxvebia 

Cveulebrivi saangariSo: sams mivumatoT rva; aTs gamovakloT 

oTxi da ase Semdeg _ TxuTmetis farglebSi. 

218/9/! kanonzomierebaa: eqvskuTxa Sekruli texili, 

SvidkuTxa, rvakuTxa... matulobs zeda qimebis raodenoba. 

g 108. {fqjsj!Tflsfcb/!yb{fcjt!ebkhvgfcb/!! 

219/2/! {fqjsj! Tflsfcb: 10+6, 6+10, 9+7, 7+9, 8+8.! yvelaferi 

gakeTdeba iseve, rogorc 93.1-Si, oRond TeqvsmetisaTvis. 

219/3/! psj! qbsbmfmvsj! hbnplmfcb! tbzpgbdypwsfcp! 

bnpdbofcTj-! spdb! npobxjmfpct! ovmj/! iseve, rogorc 106.2-Si, 

SekiTxvebiT damuSavdeba da amoixsneba ori amocana. pirveli _ 

dafaze, meore _ mxolod msr-Si (108.2). aqac mcire siaxlea 

sityva àrcerTi~. mis Sesabamisad unda Caiweros 0. 

1) bavSvs hyavs 8 biZaSvili da 5 deidaSvili. wels dabadebis 

dReze aklda 3 biZaSvili, deidaSvili ki yvela iyo, arcerTi 

ar klebia. ramdeni biZaSvili da ramdeni deidaSvili yofila 

dabadebis dReze? 

2) dafaze ewera 7 cali sityva da 5 cali cifri. masw.-ma 

waSala 7 sityva, xolo cifrebidan arcerTi ar wauSlia, 

piriqiT, kidev 4 cifri dawera. ramdeni sityva da ramdeni 

cifri gaxda dafaze? 

meore amocanaSi mcire siaxlea: gamoklebasTan erTad 

paraleluri mimatebacaa. 

219/4/! UwbmtbDjope! npdfnvmj! yb{fcjt! ebkhvgfcb;! npoblwf. 

Ufcj-! Tflsvmj! ufyjmfcj-! hbytojmj! ufyjmfcj-! Tflsvmj! nsv.

- 278 - 

efcj-!hbytojmj!nsvefcj/!! jer swrafad gavameorebinebT mTw-s 

geometriul TvalsaCinoebaSi xazebis dajgufebas: monakveTebi, 

Sekruli texilebi, gaxsnili texilebi, Sekruli mrudebi, gaxsnili 

mrudebi, sxvebi. ganvixilavT mxolod xazebis jgufebs. 

Semdeg Seivseba cxrili (108.3). mis TviTeul ujredSi 

Caixazeba Sesabamisi xazi (araferi ar Caiwereba). 

219/5/!`txsbgj!qbtvyfcj!`!sjdywjt!eblblvofcb~/ ix. 102.4. 

219/9. gairCes iseve, rogorc 106.8. aq unda exazos: 

_.. /... |... |.... \.... _.... 

g 109. {fqjsj!hbnplmfcb/!Ubwj!22/3.jt!eb!21/4.jt!hbnfpsfcb/!!! 

21:/2/! {fqjsj! hbnplmfcb;! 27`7-! 27`8-! ///! -! 27`21/! yvelaferi 

gakeTdeba iseve, rogorc 94.1-Si, oRond TeqvsmetisaTvis. 

21:/3/! Ubwj! 22/3.jt! hbnfpsfcb! eb! Tfkbnfcb/! SekiTxvebiT damu- 

Savdeba da amoixsneba ori amocana, sqemebi _ msr-Sia (109.2). 

1) guSinwin Taroze 4 wigni da 6 rveuli ido. guSin 

daumates 2 wigni, samagierod, 3 rveuli moakles. 

ramdeni wigni da ramdeni rveuli gaxda Taroze guSin? 

am amocanis CarCoSi `Cawerisas~ mcire siaxlea: CarCoSi 

Caiwereba 4 da 6, marjvniv ki miewereba +2 da –3. Semdeg kidev 

marjvniv Camoismeba xazi da miewereba pasuxebi. 

2) moswavles unda daewera 5 cifri da unda amoexsna 4 

amocana. moswavlem dawera 4 cifri, xolo amocanebidan 

arcerTi ar amouxsnia. ramdeni cifri darCa mas dasaweri da 

ramdeni amocana darCa mas amosaxsneli? 

21:/4/! Ubwj! 21/4.jt! Tfkbnfcb! eb! hbnfpsfcb/! ! ! gaizomeba skamis 

sigane. Sedegi dafaze daiwereba {2 dm}. 

_ Tu iciT, zangebi rogori xalxia? {Savkanianebi} isini 

cxovroben Sors, afrikaSi. afrikaSi spiloebi, Jirafebi, 

behemotebi da mravali sxva didroni cxoveli binadrobs. 

afrika sul sxva qveyanaa. afrikis skolaSic saxazavebiT 

zomaven manZilebs. warmoidgineT, rom es Cveni skami waiRes 

afrikaSi da iq gazomes iqauri saxazaviT. rogor ggoniaT, ra 

sigrZes miiRebdnen? {imave 2 decimetrs} saidan ici? {yvela

- 279 - 

saxazavis decimetrebi tolia, sulerTia, vin zomavs, yvela 

decimetri zustad tolia}. 

Semdeg CavatarebT sxva gazomvebs saklaso saxazaviT. jer 

ramdenime moswavles erTdroulad gavazomvinebT manZilebs 

(magaliTad, fanjarasa da skams Soris) _ sigrZeebi unda 

miviRoT araumetes 9 decimetrisa da 9 santimetrisa. yvela 

Sedegi dafaze daiwereba. 

Semdeg gaizomeba SedarebiT didi manZili, sigrZe iyos 16 

decimetramde, oRond Sedegi ar daiwereba. 

masw.-ma yuradReba unda miaqcios, rom ukanaskneli ramdenime 

gakveTilis ganmavlobaSi yovel moswavles TiTojer mainc 

mouwios gazomvis Catarebam sakuTari xeliT. 

21:/5/!`txsbgj!qbtvyfcj!`!sbnefoj!UjUjb~/! ix. 52.4, oRond, 

TiTebis raodenoba aRwevs Teqvsmetamde. amitom, cxadia, maswavlebels 

mouwevs raodenobis or wyebad Cveneba (magaliTad, jer 

aTi da Semdeg oTxi). 

21:/7/ amgvari amocanebi awi Zalian xSirad iqneba. amitom 

saWiroa axsna imisa, Tu risi gakeTeba moeTxovebaT. klasSi 

Sesruldeba is moqmedebani, romlebic amocanis pirobaSia 

aRwerili. Sedegi daiwereba dafaze masw.-is mier. _ Sinac ase 

unda gaakeToT, oRond pasuxs aqve CawerT (masw.-i miuTiTebs 

patara badian CarCoze)! Tavidan amgvari amocanebis amoxsna zog 

moswavles SeiZleba gauWirdes, magram sakmaod male iswavlian 

da aRar gauWirdebaT! 

21:/8-Si kanonzomierebaa: cxrilis yovel svetSi da yovel 

striqonSi unda iyos TiTo-TiTo lurji samkuTxedi, Savi wre 

da ruxi marTkuTxedi. amgvari cxrilebi klass ukve Seuvsia _ 

ix. 48.3, 48.6, 50.8 da sxva. SevaxsenoT es Zveli cxrilebi da 

maTi Sevsebis kanonzomiereba. magram 109.7-Si erTi siaxlea: 

wreebic, marTkuTxedebica da samkuTxedebic araa formiTa da 

zomiT erTnairebi. es damatebiTi siZnelea, radganac mcire ganzogadebas 

moiTxovs. cxrilSi unda exazos: qveda marcxena ujredSi 

_ raime ruxi marTkuTxedi (sulerTia, rogori!), zeda 

marjvena ujredSi _ raime Savi wre (sulerTia, ra zomisa!).

- 280 - 

21:/9-s magvarebi ukve ramdenjerme iyo: 94.7, 95.8, 104.8... 

g 110. `bUfvmj!eb!Twjej~-!tbzpg/!bnpdbo-!oblwU/!ebkhvg/!! 

221/1/ 109.6-Si Tuki vinmes mcdari pasuxi eqneba, uceb klas- 

Sive CxirebiT gadaTvalos da pasuxi Seasworos. 

TfojTwob/ SemdgomSi amgvar amocanebs amas aRar mivuwerT, Tumca 

Secdomisas yovelTvis saWiro iqneba klasSi CxirebiT gadaTvla. 

221/2/! Dwjenfujbo! hspwbTj! bUfvmjt! hbnpzpgb/! kvgUfcjU! 

ebUwmb;! `fsUj! dbmj! bUfvmj! eb! obTUj! Twjej~/! nplmfe;! 

`bUfvmj! eb! Twjej~-! bov;! `bUj! eb! Twjej~/! `sbnefojU! nfujb! 

bU{f@~! `!`TwjejU! nfuj~/! hbb{sfcb! tjuzwjtb! Ù.Twje.nfuj~/!! 

Dwjenfujt! Tfebsfcb! obtxbwm! sjdywfcUbo/! yvelaferi 

gakeTdeba iseve, rogorc 92.1-Si, oRond CvidmetisaTvis. 

------------------------------------------------------------------------------------ 

II-12. psj! Ubonjnefwsvmj! Tflsfcb! bo! psj! hbnplmfcb! 

tbzpgbdypwsfcp! bnpdbofcTj

- 281 - 

2) zazam xval 3 simRera unda imReros da 4 leqsi unda 

Tqvas. zeg simRera ar unda imReros, leqsi ki imdenive unda 

Tqvas, ramdenic xval. sul ramdeni simRera unda imReros da 

ramdeni leqsi unda Tqvas zazam? 

------------------------------------------------------------------------------------ 

III-11. ebkhvgfcb! hpofcbTj! xbsnpehfojU/! ori TvalsaCino 

niSanTvisebis mixedviT ramdenime jgufad dajgufdeba kargad 

cnobili sagnebi, oRond maTi mxolod sityvieri dasaxelebisa da 

dafaze Camoweris safuZvelze. dajgufebis Sedegebic Camoiwereba 

dafaze, kvlav sityvierad (cxadia, am droisaTvis klasma ukve 

sakmaod kargad unda icodes kiTxva). bolos igive keTdeba 

maTematikuri sagnebisaTvis. 

ori TvalsaCino niSanTvisebis mixedviT dajgufebani klass 

adreve Sexvedria (ix., mag., 21.3, 22.3, 22.6, 24.3, 25.3...). magram 

aqamde dajgufeba (da, Sesabamisad, faruli ganzogadebac) 

emyareboda uSualod mocemul suraTebs. axla siaxlea is, rom 

mocemulia mxolod sityvebi. es arsebiTad arTulebs gaazrebas. 

bavSvi erTmaneTisagan ver arCevs sityvas, mis mniSnelobasa da 

Sesatyvis sagans. amitom veubnebiT `davajgufoT sityvebi~ da amiT 

vajgufebinebT gonebaSi warmodgenil Sesatyvis sagnebs, ufro 

zustad, maT xatebs. amasTan, araa saWiro Tqma `cxovelebis 

aRmniSvneli sityvebi~. es Zalian rTulia. vityviT martivad: 

`cxovelebi~. sityva da misi Sesatyvisi sagani aRreulia da 

eqvswliani bavSvisaTvis es gardauvalia! 

SeniSvna: pirvelklasel bavSvs wesierad ar esmis sulierisa da 

usulos gansxvaveba, aseve cocxalisa da aracocxalis gansxvaveba. es 

cnebebi mas arc sWirdeba! Cven am cnebebs vxmarobT mxolod masw.isTvis, 

da ara moswavlisTvis. 

garda amisa, viTardeba gazomvis Temac, ganmtkicdeba ejej! 

tbyb{bwjt!tlbmjt (cxadia, am terminis gareSe) codna. 

------------------------------------------------------------------------------------ 

221/4/! tjuzwjfsbe! npdfnvmj! ojwUfcjt-! tvmjfsfcjtb! eb! 

oblwUfcjt! ebkhvgfcb! sbnefojnf! khvgbe! fsUj! UwbmtbDjop! 

ojTboUwjtfcjt! njyfewjU/! ! jer CavatarebT ramdenime gazomvas

- 282 - 

saklaso saxazaviT. gaizomeba SedarebiT didi manZilebi, sigrZe 

iyos 17 decimetramde, oRond Sedegi ar daiwereba. 

masw.-ma yuradReba unda miaqcios, rom ukanaskneli ramdenime 

gakveTilis ganmavlobaSi yovel moswavles TiTojer mainc 

mouwios gazomvis Catareba sakuTari xeliT. 

Semdeg masw.-i dafaze erT svetSi Camowers sityvebs: 

vaSli, magida, gvirila, msxali, Sesakrebi, skami, atami, 

sxvaoba, vardi, naSTi, sawoli, ia, bali, iasamani, saklebi, 

jami, leRvi, maklebi, komSi, Werami, yayaCo, karada. 

_ radara jgufebad SeiZleba davajgufoT es sityvebi? 

{yvavilebi, xili, maTematikuri sityvebi, aveji} davweroT! 

masw.-i gverdiT maRla dawers: yvavilebi da 

Camousvams xazs. imave striqonSi moSorebiT dawers 

xili da Camousvams xazs. imave striqonSi moSorebiT 

dawers aveji da Camousvams xazs. imave striqonSi moSorebiT 

dawers maT. sityvebi da Camousvams xazs. 

Semdeg moswavleebs zepirad aTqmevinebs (waakiTxebs), Tu 

romeli sityvebi Sedis amaTuim erT jgufSi. TviTeuli 

dasaxelebis qveS erT svetSi Camoiwereba Sesabamisi sityvebi 

mocemuli siidan. moswavleebi Tavad daajgufeben. bolos, masw.i 

TviTeuli jgufisaTvis kidev or-or sityvas daamatebinebs 

(dafaze sityvebs mxolod masw.-i wers, moswavleebi zepirad 

hkarnaxoben). 


saangariSo: sams mivumatoT rva; aTs gamovakloT oTxi da ase 

Semdeg _ Teqvsmetis farglebSi. 

221/8. erTi kanonzomiereba pirdapir Cans: 1, 2, 3, 4. 

amasTan, SedarebiT misdagvar amocanebTan (magaliTad, 108.8), 

erTi siaxlea. es igive siaxlea, rac iyo 109.7-Si: erTi jgufis 

nakvTebi erTmaneTisagan gansxvavdeba formiTa da zomiT. amitom 

saWiro xdeba mcire ganzogadeba: sulerTia, rogori samkuTxedia,

- 283 - 

pirvel striqonSi sul samkuTxedebia, meoreSi _ oTxkuTxedebi, 

mesameSi _ wreebi, meoTxeSi _ monakveTebi. 

g 111. {fqjsj!Tflsfcb/!eje!tbyb{bw{f!nbo[jmjt!hbslwfwb/!!! 

222/2/! {fqjsj! Tflsfcb: 10+7, 7+10, 9+8, 8+9.! yvelaferi 

gakeTdeba iseve, rogorc 93.1-Si, oRond CvidmetisaTvis. 

222/3/!psj!qbsbmfmvsj!hbnplmfcb!tbzpgbdypwsfcp!bnpdbof. 

cTj-!spdb!npobxjmfpct!tjuzwb!`jnefojwf~!eb-!Tftb[mpb-!ovmj/ 

es, arsebiTad, mxolod gameoreba da ganmtkicebaa naswavlisa. 

iseve, rogorc 110.2-Si, SekiTxvebiT damuSavdeba da 

amoixsneba ori amocana. pirveli _ dafaze da Tan msr-Si, 

meore _ mxolod msr-Si (111.2). 

1) moswavles unda daexaza 8 marTkuTxedi da amdenive 

samkuTxedi. mas dasaxazavi darCa 3 marTkuTxedi da amdenive 

samkuTxedi. ramdeni marTkuTxedi da ramdeni samkuTxedi 

dauxazavs moswavles? 

2) xeze isxdnen Citebi: 9 mercxali da 5 beRura. cota 

xanSi ramdenime Citi gafrinda. xeze darCa 4 mercxali da 

amdenive beRura. ramdeni mercxali da ramdeni beRura 

gafrenila? 

222/4/! eje! tbyb{bw{f! UjUfcjU! npojTovm! behjmfct! Tpsjt! 

nbo[jmjt! hbslwfwb! `! spdb! nbo[jmjb! sbnefojnf! en! vobTUpe/!!! 

masw.-i saklaso metrian saxazavs magidaze ise dadebs, rom 

klasi mis skalas xedavdes (Tuki Zalian didi klasia, saxazavi 

xelSi daikaos da merxebs Soris Camoataros). masw.-i ori Ti- 

TiT moniSnavs saxazavze adgilebs. moswavleebma unda daTvalon, 

am TiTebs Soris ramdeni decimetria (naSTi ar rCeba). 

Tavidan TiTebis mdebareoba emTxveva did danayofebs, magali- 

Tad, 2 dm (an 20 sm _ saxazavs gaaCnia) da 7 dm (70 sm). moswavleebi 

warwerebs yuradRebas ar aqceven, ubralod iTvlian, 

ramdeni decimetria TiTebs Soris {xuTi}. Sesruldeba mxolod 

ramdenime amgvari savarjiSo. Semdeg TiTebis mdebareoba 

emTxveva `naxevrebs~, magaliTad, 35 sm da 95 sm {maT Soris 

manZilia 6 dm}. moswavles mivaxvedrebT, rom TiTo-TiTo deci-

- 284 - 

metrebi am SemTxvevaSic gadaiTvleba. 

Semdeg masw.-i dafaze Camoxazavs mokle Sveul monakveTebs, 

miadebs maT saxazavs da klass daaTvlevinebs maT Soris manZils 

(orive monakveTi did danayofebs unda daemTxves!). Sesruldeba 

mxolod ramdenime amgvari savarjiSo (danarCenebi _ TamaSSi). 

Sesruldeba agreTve ramdenime Sebrunebuli savarjiSo: 

_ Seni TiTebiT maCvene am saxazavze 4 dm manZili! 

222/5/!`txsbgj!qbtvyfcj~!`!sb!nbo[jmjb!UjUfct!Tpsjt/ grZel 

deba 111.3-is SekiTxvebi: pirvel wreze _ advilebi (did danayofebs 

Soris), momdevno wreebze _ Znelebi (`naxevrebs~ Soris). 

222/9. SedarebiT 109.7-Tan, gaZnelebulia. jer erTi, 

moswavles aRar exmareba nakvTebis feri, yvela nakvTi erTi 

ferisaa. meorec, ganzogadeba ufro metadaa saWiro: moswavlem 

unda dainaxos saerTo Tvisebebi: cxrilSia samkuTxedebi, 

oTxkuTxedebi da monakveTebi. rogorc yvela amgvar cxrilSi _ 

TiTo svetsa da TiTo striqonSi _ TiTo-TiTo. amitom aklia 

ori monakveTi. 

g 112. {fqjsj!hbnplm/!nbo[jmjt!eblbwTjsfcb!tywbpcbtUbo. 

223/2/! {fqjsj! hbnplmfcb;! 28`8-! 28`9-! 28`:-! 28`21/! 


CvidmetisaTvis. 

223/3/! psj! Ubonjnefwsvmj! Tflsfcb! tbzpgbdypwsfcp! bnpdb. 

ofcTj/! SekiTxvebiT damuSavdeba da amoixsneba ori amocana. 

pirveli _ dafaze da Tan msr-Si, meore _ mxolod msr-Si 

(112.2). am gakveTilis amocanebi Zalian advilebia, imdenad, rom, 

arsebiTad, arcaa ormoqmedebiani. erTi kia _ Tan vameorebinebT 

Znel droiT mimarTebebs. 

1) guSinwin moswavlem 3 leqsi waikiTxa, guSin _ 2 leqsi, 

dRes _ 3 leqsi. sul ramdeni leqsi waukiTxavs moswavles? 

2) muSebma xval 4 ormo unda gaTxaron, zeg _ 2 ormo, 

mazeg _ 3 ormo. sul ramdeni ormo unda gaTxaron muSebma? 

am amocanebis sqema sxvagvaria (ix. msr 112.2). gavarCioT, 

magaliTad, pirveli amocana.

- 285 - 

_ ramdeni leqsi waikiTxa moswavlem guSinwin? {3} 

CavweroT aq (Cawers marcxena ujredSi). guSin? {2} sad 

Cavwero 2-iani? {gverdiT, Sua ujredSi} CavweroT! dRes 

ramdeni waikiTxa? {3} sad ra Cavwero? {marjvena ujredSi 3iani} 

CavweroT! amocana ras gvekiTxeba? {sul ramdeni leqsi 

waukiTxavs moswavles?} levan, ucqire am naxazs da srulad 

waikiTxe amocanis piroba! kargi, axla amovxsnaT amocana. ra 

moqmedeba unda CavataroT? {Sekreba} vin iangariSebs? (SeiZleba 

ricxvTa kibec gamoiyenon, rogorc urCevniaT). ra aris 

amocanis pasuxi? {8 leqsi} sad Cavwero? {qvemoT, `jamSi~}. 


nbo[jmjt! eblbwTjsfcb! sjdywUb! tywbpcbtUbo! `! spdb! nbo[jmjb! 

sbnefojnf! en! vobTUpe. yvelaferi gakeTdeba iseve, rogorc 

111.3-Si, oRond orive TiTi yovelTvis did danayofebs 

davamTxvioT, `naxevrebi~ aRar gvinda. 

_ maS, ramdeni dm yofila manZili am TiTebs Soris? {8 dm} 

kargi, magram amas dauTvleladac SeiZleboda mivmxvdariyavT! 

marTlac, aba waikiTxeT, ra cifri weria am TiTTan? {1-iani} 

meore TiTTan? {9-iani} ramdeni aklia 1-s 9-mde? gavixsenoT: 

erT ricxvs meoremde aklia imdeni, ramdenicaa... ra? {am 

ricxvebis sxvaoba} diax, ramdenia cxrisa da erTis sxvaoba? 

{8} hoda swored es 8 dm aris manZili. maS, TiTebs Soris 

manZili pirdapir SegveZlo gagvego. amisaTvis cxrisTvis unda 

gamogveklo erTi. aba, axla swrafad gamoTvaleT, ramdeni 

decimetria TiTebs Soris manZili? 

Semdeg masw.-i dafaze Camoxazavs mokle Sveul monakveTebs, 

miadebs maT saxazavs da klass gamoaTvlevinebs maT Soris 

manZils (orive monakveTi did danayofebs unda daemTxves!). 


_ Seni TiTebiT maCvene am saxazavze 4 dm manZili! 

223/5/!igive, rac 111.4, oRond santimetriani naSTebiT. 

223/8 hgavs 111.8-s, mcired gaZnelebulia. cxrilSia: wre,

- 286 - 

xuTkuTxedi, oTxkuTxedi (es ukanaskneli yvelaze Znelia) _ 

TiTo striqonsa da TiTo svetSi zustad TiTo-TiToa. amitom 

qveda marcxena ujredSi unda exazos raime oTxkuTxedi 

(sulerTia, rogori), Sua ujredSi _ raime wre. 

g 113. `bUfvmj!eb!swb~-!vnbsujw/!psnprnfe/!tbzpg/!bnpdbo/ 

224/2/! Uwsbnfujbo! hspwbTj! bUfvmjt! hbnpzpgb/! kvgUfcjU! 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!swb~/!nplmfe;!`bUfvmj! 

eb! swb~-! bov;! `bUj! eb! swb~/! `sbnefojU! nfujb! bU{f@~!`!`swjU! 

nfuj~/! hbb{sfcb! tjuzwjtb! Ù.swb.nfuj~/! Uwsbnfujt! Tfebsfcb! 

obtxbwm!sjdywfcUbo/ yvelaferi gakeTdeba iseve, rogorc 92.1- 

Si, oRond TvrametisaTvis. 

224/3/!vnbsujwftj!psnprnfefcjboj!tbzpgbdypwsfcp!bnpdbofcj/ 

SekiTxvebiT damuSavdeba da amoixsneba ori amocana. pirveli 

_ dafaze da Tan msr-Si, meore _ mxolod msr-Si (113.2). 

1) SarSanwin dedaRors 3 goWi hyavda, SarSan _ 2 goWiT 

meti, wels _ 3 goWiT meti, vidre SarSan. ramdeni goWi hyavs 

dedaRors wels? 

2) durgalma zafxulSi 3 skami gaakeTa, Semodgomaze _ 2 

skamiT meti, zamTarSi _ 3 skamiT meti, vidre Semodgomaze. 

ramdeni skami gaukeTebia durgals zamTarSi? 

es amocanebi sxvagvarad Caiwereba (msr 113.2). gavarCioT, 

magaliTad, pirveli amocana: _ ramdeni goWi hyavda SarSanwin? 

{sami} ai aq weria `SarSanwin~ da mivuweroT 3 g. ras aRniSnavs 

aso g? {goWs} kargi, SarSan ramdeniT meti hyavdao? {oriT} ra 

moqmedeba unda SevasruloT, raTa gavigoT, SarSan ramdeni 

hyolia? {Sekreba} ai aq weria `SarSan~ da mivuweroT: 3 g + 2 

g. wels ra moxdao? {kidev samiT meti goWi hyavdao} ra 

moqmedeba unda SevasruloT, raTa gavigoT, wels ramdeni hyolia? 

{Sekreba} ai aq weria `wels~ da mivuweroT: 3 g + 2 g + 3 g. 

saboloo sqema amgvaria: SarSanwin _ 3 g 

SarSan _ 3 g + 2 g 

wels _ 3 g + 2 g + 3 g

- 287 - 

_ naTia, ucqire am naxazs da srulad waikiTxe amocanis 

piroba! kargi, axla amovxsnaT amocana. SarSan ramdeni goWi 

eyoleboda? (da masw.-i Tan TiTs daadebs Sesabamis Canawers 

`3g + 2g~) {3 + 2 = 5 _ miewereba sityvis gaswvriv}. wels 

ramdeni goWi eyoleboda? (da masw.-i Tan TiTs daadebs Sesabamis 

Canawers `3 g + 2 g + 3 g~) {5 + 3 = 8 _ miewereba sityvis 

gaswvriv}. kidev rame unda SevasruloT? {ara, amocana 

amoxsnilia} vin ityvis srul pasuxs? {dedaRors wels hyolia 

rva goWi}. 

224/4/!eje!tbyb{bw{f!UjUfcjU!npojTovm!behjmfct!Tpsjt!nbo. 

[jmjt!hbslwfwb!`!spdb!nbo[jmjb!sbnefojnf!en!eb!sbnefojnf!tn/ 

yvelaferi gakeTdeba iseve, rogorc 111.3-Si, oRond TiTebsa Tu 

dafaze daxazul monakveTebs Soris manZili meore da Semdgom 

savarjiSoebSi iqneba naSTiani: 7 dm 5 sm; 6 dm 8 sm da sxva. 

es sigrZeebi dafazec daiwereba. oRond, pirveli TiTi 

yovelTvis did danayofs davamTxvioT, `naxevrebi~ aRar gvinda. 

Tanafardoba 1 dm = 10 sm ar gviswavlebia! 


_ Seni TiTebiT maCvene am saxazavze 4 dm da 7 sm manZili! 

224/5/ igive, rac 111.4, oRond pasuxi unda Tqvan ufro 

swrafad _ gamoklebis gamoyenebiT. 

224/8/ cxrilSia: monakveTebi, gaxsnili texilebi da Sekruli 

mrudebi. cariel ujredebSi unda exazos: raime gaxsnili 

texili (sulerTia, rogori) da raime Sekruli mrudi 

(sulerTia, rogori). 

g 114. {fqjsj!Tflsfcb/!Ubonjnefw/!hbnplm/!oblwUfcjt!ebkhvg/ 

225/2/!{fqjsj!Tflsfcb: 10+8, 8+10, 9+9.! yvelaferi gakeTdeba 

iseve, rogorc 93.1-Si, oRond TvrametisaTvis. 

225/3/! psj! Ubonjnefwsvmj! hbnplmfcb! tbzpgbdypwsfcp! 

bnpdbofcTj/! ori amocana damuSavdeba imdagvarive SekiTxvebiT, 

msjelobiTa da sqemis SevsebiT, rogorebic iyo 113.2-Si. 

cxadia, sqemebSi mimateba da + niSani Seicvleba Sesabamisad 

gamoklebiTa da _ niSniT. pirveli amocana amoixsneba dafaze

- 288 - 

da Tan msr-Si, meore _ mxolod msr-Si. 

1) guSinwin 8 moswavle iyo avad, guSin _ 2-iT naklebi, dRes 

_ 4-iT naklebi, vidre guSin. ramdeni moswavlea avad dRes? 

2) vazs 9 mtevani yurZeni esxa. bebom mowyvita 3 mtevani, 

bavSvebmac _ amdenive. ramdeni mtevani yurZeni darCenila vazze? 

225/4/! ! tjuzwjfsbe! npdfnvmj! ojwUfcjt-! tvmjfsfcjtb! eb! 

oblwUfcjt! ebkhvgfcb! sbnefojnf! khvgbe! psj! UwbmtbDjop! 

ojTboUwjtfcjt! njyfewjU/! jer swrafad gakeTdeba ramdenime 

savarjiSo TiTebs an daxazul monakveTebs Soris manZilis 

gamoTvlaze (rogorc 113.3-Si). swrafadve Sesruldeba agreTve 

ramdenime Sebrunebuli savarjiSo: _ Seni TiTebiT maCvene am 

saxazavze 4 dm da 7 sm manZili! 

Semdeg masw.-i dafaze erT svetSi Camowers sityvebs (Tan 

moswavleebi waikiTxaven): kata, kovzi, daTvi, monakveTi, 

marTkuTxedi, ZaRli, culi, Zroxa, kvadrati, tura, 

samkuTxedi, mela, texili, TefSi, CaquCi, tafa, Txa, niCabi, 

ciyvi, rvakuTxedi, mrudi. 

_ radara jgufebad SeiZleba davajgufoT es sityvebi? 

{nakvTebi, cxovelebi da nivTebi} davweroT! 

masw.-i gverdiT maRla dawers: nakvTebi da Camousvams 

xazs. imave striqonSi moSorebiT dawers cxovelebi da 

Camousvams xazs. imave striqonSi moSorebiT dawers nivTebi 

da Camousvams xazs. Semdeg mets jer araferi ar dawers, 

moswavleebs zepirad aTqmevinebs (waakiTxebs), Tu romeli 

sityvebi Sedis erT jgufSi da romlebi _ meoreSi. 

_ magram yvela cxoveli erTnairi araa! cxovelebic 

SeiZleba or jgufad gavyoT. radara jgufis cxovelebia aq? 

{Sinaurebi da gareulebi} davweroT! arc yvela nivTia 

erTnairi. nivTebic SeiZleba or jgufad gavyoT. radara 

jgufis nivTebia aq? {saojaxo nivTebi da samuSao nivTebi} 

davweroT! nakvTebi radara jgufebad SeiZleba gavyoT?

{mravalkuTxedebi da xazebi}. 

dafaze amgvari sqema iqneba: 

- 289 - 

Semdeg TviTeuli dasaxelebis qveS erT svetSi Camoiwereba 

Sesabamisi sityvebi mocemuli siidan. moswavleebi Tavad 

daajgufeben. bolos, masw.-i TviTeuli jgufisaTvis kidev oror 

sityvas daamatebinebs (dafaze sityvebs mxolod masw.-i 

wers, moswavleebi zepirad hkarnaxoben). 

225/5/!`txsbgj!qbtvyfcj!`!UjUfcjt!Dwfofcb~/ ix. 97.4, oRond 

raodenoba aRwevs Tvrametamde. 

225/8/ cxrilSia: Sekruli texilebi, gaxsnili texilebi da 

monakveTebi (rogorc yvela amgvar cxrilSi _ TiTo svetsa da 

TiTo striqonSi _ TiTo). cariel ujredSi unda exazos raime 

monakveTi (sulerTia, rogori). 

g 115. {fqjsj!hbnplmfcb/!eje!tbyb{bw{f!nbo[jmjt!hbslwfwb/! 

226/2/! {fqjsj! hbnplmfcb;! 29`9-! 29`:-! 29`21/! yvelaferi 

gakeTdeba iseve, rogorc 94.1-Si, oRond TvrametisaTvis. 

226/3/! psnprnfefcjboj! tbzpgbdypwsfcp! bnpdbofcj-! spnfmUb! 

bnpytob! npjUypwt! sjdywjtb! eb! kbnjt! Tflsfcbt/! ! SekiTxvebiT 

damuSavdeba da amoixsneba ori amocana. pirveli _ dafaze da 

Tan msr-Si, meore _ mxolod msr-Si (115.2). 

1) bebos 3 beWedi aqvs, mamidas _ 1-iT meti. ramdeni beWedi 

aqvT orives erTad? 

2) wina kviraSi 2 wvimiani dRe iyo, am kviraSi _ 3-iT 

meti. ramdeni wvimiani dRe yofila orive kviraSi erTad? 

amocanebis sqema sxvagvaria (ix. msr), dafazec aseTi daixazeba 

pirveli amocanis amoxsnisas. masw.-i dafaze Seavsebs 

moswavleTa pasuxebis safuZvelze, moswavleebi _ msr-Si. 

marcxena CarCoSi iwereba pirveli monacemi (3 b.). 

_ romeli moqmedeba unda SevasruloT, raTa gavigoT, ramdeni

- 290 - 

beWedi aqvs mamidas? {mimateba} ai am marjvena CarCoSi 

CavweroT 3 b + 1 b. es ras gviCvenebs? {ramdeni beWedi 

aqvs bebos} es? {ramdeni beWedi aqvs mamidas} aq, `jamSi~, ra 

unda CavweroT? {ramdeni beWedi aqvT orives erTad} lela, 

ucqire am naxazs da srulad waikiTxe amocanis piroba! kargi, 

axla amovxsnaT amocana. ra moqmedeba unda SevasruloT? 

{Sekreba} (masw.-i CarCoebs Soris dawers did + niSans) ramdeni 

beWedi hqonia mamidas? {oTxi} orives erTad? 

qveda `jamSi~ (elifsSi) Caiwereba meore Sekrebis Sedegad 

miRebuli ricxvi. bolos, gairkveva, romelia amocanis pasuxi. 


nbo[jmjt!hbslwfwb!`!spdb!nbo[jmjb!sbnefojnf!en!eb!sbnefojnf! 

tn-!bo!sbnefojnf!en-!bo!sbnefojnf!tn/!! 

yvelaferi gakeTdeba iseve, rogorc 113.3-Si, oRond TiTebsa 

Tu dafaze daxazul monakveTebs Soris manZili iqneba: xan 

unaSTo dm-ebi (5 dm, 2 dm), xan _ dm-ebi da sm-ebi (7 dm 5 sm; 

6 dm 8 sm) da xan _ mxolod sm-ebi (9 sm, 3 sm, 1 sm da 

sxva). es sigrZeebi dafazec daiwereba. pirveli TiTi yovel- 

Tvis did danayofs davamTxvioT, `naxevrebi~ aRar gvinda. 


_ Seni TiTebiT maCvene am saxazavze 4 dm da 7 sm manZili! 

8 dm manZili! 9 sm manZili! (da sxva). 


saangariSo: sams mivumatoT rva; aTs gamovakloT oTxi da ase 

Semdeg _ Tvrametis farglebSi. 

226/8/ cxrilSia: Sekruli mrudebi, gaxsnili mrudebi da 

wertilebi. cariel ujredebSi unda exazos: wertili da raime 

gaxsnili mrudi (sulerTia, rogori). 

g 116. `bUfvmj!eb!dysb~/!psnprn/!tbz/!bnpdbofcj/!hbnfpsfcb/!! 

227/2/! dysbnfujbo! hspwbTj! bUfvmjt! hbnpzpgb/! kvgUfcjU! 

ebUwmb;!`fsUj!dbmj!bUfvmj!eb!obTUj!dysb~/!nplmfe;!`bUfvmj! 

eb! dysb~-! bov;! `bUj! eb! dysb~/! `sbnefojU! nfujb! bU{f@~! `! 

`dysjU! nfuj~/! hbb{sfcb! tjuzwjtb! `dysb.nfuj~/! dysbnfujt!

- 291 - 


rogorc 92.1-Si, oRond cxrametisaTvis. 

227/3/! psnprnfefcjboj! tbzpgbdypwsfcp! bnpdbofcj-! spnfmUb! 

bnpytob!npjUypwt!sjdywjtb!eb!tywbpcjt!Tflsfcbt/!es sakiTxi 115.2is 

magvaria, oRond `-iT metia~ Secvlilia `-iT naklebia~-Ti. amis 

Sesabamisad sqemaSi `-iT meti~ Seicvleba `-iT naklebiT~. 

SekiTxvebiT damuSavdeba da amoixsneba ori amocana. pirveli 

_ dafaze da Tan msr-Si, meore _ mxolod msr-Si (116.2). 

1) wiklaurebis ojaxSi 7 sulia, WinWaraulebis ojaxSi _ 

4-iT naklebi. ramdeni sulia orive ojaxSi erTad? 

2) wels skolas 5 klasi daamTavrebs, gaisad _ 2-iT naklebi. 

ramdeni klasi daamTavrebs skolas am ori wlis ganmavlobaSi? 

masw.-i sqemas dafaze daxazavs da Seavsebs moswavleTa 

pasuxebis safuZvelze, moswavleebi _ msr-Si. marcxena CarCoSi 

iwereba pirveli monacemi. 

_ romeli moqmedeba unda SevasruloT, raTa gavigoT, ramdeni 

sulia WinWaraulebis ojaxSi? {gamokleba} ai am marjvena 

CarCoSi CavweroT 7s _ 4s. es ras gviCvenebs? {ramdeni sulia 

wiklaurebis ojaxSi} es? {ramdeni sulia WinWaraulebis 

ojaxSi} aq, `jamSi~, ra unda CavweroT? {ramdeni sulia orive 

ojaxSi erTad} nana, ucqire am naxazs da srulad waikiTxe 

amocanis piroba! kargi, axla amovxsnaT amocana. ramdeni suli 

yofila WinWaraulebis ojaxSi? {sami} axla ra moqmedeba unda 

SevasruloT? {Sekreba} (masw.-i CarCoebs Soris dawers did + 

niSans) radara unda SevkriboT? {7 + 3 = 10}. 

qveda `jamSi~ (elifsSi) Caiwereba Sekrebis Sedegad 

miRebuli ricxvi. bolos, gairkveva, romelia amocanis pasuxi. 

227/4/!Ubwj!22/4.jt!Tfkbnfcb!eb!hbnfpsfcb/!!gakeTdeba ramdenime 

savarjiSo TiTebs an daxazul monakveTebs Soris manZilis 

gamoTvlaze (rogorc 115.3-Si). 


_ Seni TiTebiT maCvene am saxazavze 4 dm da 7sm manZili! 

8 dm manZili! 9 sm manZili! (da sxva). 

masw.-ma yuradReba unda miaqcios, rom ukanaskneli ramdenime

- 292 - 

gakveTilis ganmavlobaSi yovel moswavles erTxel mainc 

mouwios manZilis gamoTvla da erTxel _ dasaxelebuli 

manZilis Cveneba. 

227/5/ igive, rac 111.4, oRond 115.3-is Sesabamisi SekiTxvebiT. 

227/8/ wiTeli feri aadvilebs kanonzomierebis aRmoCenas: 

samkuTxedi, Sekruli mrudi, gaxsnili texili da ase Semdeg. 

g 117. {fqjsj!Tflsfcb/!hbnfpsfcb/!tbyb{bwjt!ebobzpgfcj/ 

228/2/ {fqjsj!Tflsfcb: 10+9, 9+10. yvelaferi gakeTdeba iseve, 

rogorc 93.1-Si, oRond cxrametisaTvis. 

228/3/ Ubwj! 23/3.jt! hbnfpsfcb! eb! Tfkbnfcb/ SekiTxvebiT 

damuSavdeba da amoixsneba sami amocana, samive _ msr-Si (117.2). 

sqemebi Seivseba iseve, rogorc wina gakveTilze. 

1) laSdiris mTaze 4 tbaa, Jibevis mTaze _ 2-iT meti. 

ramdeni tbaa orive mTaze erTad? 

2) kvaracxeliebis ojaxs 6 Zroxa hyavs, Caligavebis ojaxs 

_ 3-iT naklebi. ramdeni Zroxa hyavs orive ojaxs erTad? 

3) gasaToxni hqondaT 7 sxvadasxva yana. wina kviraSi gaToxnes 

3 yana, am kviraSi _ 2. ramdeni yana darCeniaT gasaToxni? 

----------------------------------------------------------------------------------- 

III-12. qbubsb!tbyb{bwj/!njtj!tboujnfusfcjtb!eb!ejej!tbyb{b. 

wjt!tboujnfusfcjt!upmpcb/! yb{wb! swfvmjt! gvsdfm{f! qbubsb! 

tbyb{bwjU/! dasaxazi monakveTis sigrZe Tavidan ar aRemateba 9 sms, 

raki ariTmetikis xaziT ufro didi ricxvebis Cawera jer ar 

gviswavlebia. vaxazvinebiT mocemuli zomebis mqone kvadratebs, 

marTkuTxedebs, texilebsa da sxva. geometriaSi Semogvaqvs axleburi 

cxrilebi gadanomril monakveTTa sigrZeebisa. 

Semdeg klass vavarjiSebT rveulSi daxazuli monakveTebis 

zomvaSi, ukve 20 sm-is farglebSi. milimetrebs pirvel klasSi ar 

vaswavliT, amitom yvela sigrZe santimetrebis mTel ricxvs 

Seicavs, raTa ar morCes milimetrebiani naSTi. 

------------------------------------------------------------------------------------ 

228/4/! ejej! tbyb{bwjt! tboujnfusfcjtb! eb! qbubsb! tbyb{bwjt!

- 293 - 

tboujnfusfcjt!upmpcb/!qbubsb!tbyb{bwjt!ebobzpgfcjt!xbljUywb! 

:! tn.nef/! _ Cven viswavleT gazomva am didi saxazaviT. magram 

Tqven patara saxazavebic gaqvT. rogor ggoniaT, risi gazomvaa 

mosaxer-xebeli patara saxazaviT? {patar-patara rameebsa, 

rveulSi daxazuli rameebisa...} daakvirdiT Tqvens saxazavebs. 

rogor ggoniaT, maTze decimetrebia moniSnuli Tu 

santimetrebi? {santimetrebi, decimetri xom didia} aba, 

wavikiTxoT santimetrebis cifrebi: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 

aba, yvelam Tavis saxazavze TiTebiT maCvenos manZili 7 sm, 

saidan sadamdea! (masw.-i Camoivlis da Seamowmebs). 

_ kargi, axla moifiqreT: am didi saxazavis santimetrebia 

ufro didi Tu patara saxazavis santimetrebi? 

gamoiTqmeba sxvadasxva azri. _ gaixseneT, rom yvela 

saxazavs decimetrebi aqvs zustad toli! aqac yvelagan, 

afrikaSic, ruseTSic, yvelagan. axla isev moifiqreT: am didi 

saxazavis santimetrebia ufro didi Tu patara saxazavis 

santimetrebi? {tolia} aba, SevamowmoT! 

masw.-i didi da patara saxazavebis kideebs erTmaneTs 

miadebs, maT merxebs Soris Camoatarebs da yvelas daarwmunebs, 

rom orives santimetrebi namdvilad tolia. 

228/5/!`txsbgj!qbtvyfcj!`!sbnefoj!UjUjb~/ ix. 52.4, oRond, 

TiTebis raodenoba aRwevs cxrametamde. amitom, cxadia, 

maswavle-bels mouwevs raodenobis or wyebad Cveneba 

(magaliTad, jer aTi da Semdeg oTxi). 

228/8/ hgavs 110.7-s, magram masze ufro advilia. cariel 

ujredebSi unda eweros: < 8 = 1 _ 5 < 5 

g 118. {fqjsj!hbnplm.!npdfn/!tjhs[jt!npoblwfUjt!ebyb{wb- 

229/2/! {fqjsj! hbnplmfcb;! 2:`:-! 2:`21. yvelaferi gakeTdeba 

iseve, rogorc 94.1-Si, oRond cxrametisaTvis. 

------------------------------------------------------------------------------------ 

II-13. Tflsfcb.hbnplmfcjt! vdopcj! lpnqpofoujt! np[fcob/ es 

Zalian mniSvnelovani sakiTxebia da did safrTxesac Seicavs. saqme 

isaa, rom sakmaod advilia amgvari amocanebis amoxsnis xerxebis

- 294 - 

gauazrebeli, meqanikuri daufleba wesebis gazepirebiT: imisaTvis, 

raTa vipovoT ucnobi pirveli Sesakrebi, saWiroa jams gamovakloT 

meore Sesakrebi; imisaTvis, raTa vipovoT ucnobi maklebi, saWiroa 

saklebs gamovakloT... ... es wesebi Cvenma mterma imaxsovra! am gziT 

moswavle ar gaiazrebs ariTmetikul moqmedebaTa 

urTierTSebrunebadobas, ar gaiazrebs sakiTxis arsebas da 

meqanikurad amoxsnis amocanebs. CvenTvis amgvari ram dauSvebelia. 

imisaTvis, raTa mivaRwioT namdvil gaazrebul codnas, saWiroa 

Cveneuli meTodikis mkacri dacva. 

rogorc yovelTvis, viwyebT nivTebze moqmedebebiT. zogadi 

gegma amgvaria, ganyenebis TandaTanobiTi amaRlebiT: 

1) nivTebze moqmedebebi da Tan mTw-s TvalsaCinoeba; 

2) TvalsaCino sqemebi da Tan mTw-s TvalsaCinoeba; 

3) sayofacxovrebo amocanebi romelTa konkretuli Sinaarsi aadvilebs 

sakiTxSi garkvevas, TviTon hkarnaxobs moswavles amoxsnis gzas; 

4) `Cavifiqre ricxvi~-s tipis amocanebi; 

5) ucnobi komponentis moZebna ganyenebul viTarebaSi, Sesakrebsakleb-maklebis 

SiSveli terminebiT. oRond, am terminebSi 

mocemulia mxolod amocana, xolo misi amoxsna kvlav azrianad 

unda moxdes! 

erTi sakiTxia ucnobi komponentis moZebna sayofacxovrebo amocanaSi, 

da meore sakiTxia ucnobi komponentis moZebna ganyenebul 

ariTmetikul moqmedebaSi. meore sakiTxi ufro Znelia, radgan pirvel 

SemTxvevaSi TviT amocanaTa sayofacxovrebo Sinaarsi hkarnaxobs 

moswavles imas, Tu ra moqmedeba unda Seasrulos. 

rogorc ki bavSvs gauWirdeba, unda mivaSveloT TvalsaCino 

sqema, Tuki arc es eyo kvlav pirveli safexuris nivTebze 

moqmedebani. magram mTavaria, rom ukanasknel mexuTe safexurzec ki 

araa saWiro wesebis damaxsovreba da gazepireba. sazogadod, wesebis 

Camoyalibeba araa saWiro. pirvelklaselisaTvis es naadrevia da 

fuWi. moswavle qvecnobierad daeufleba da gamoiyenebs am wesebs, 

Tanac mudmivad gacnobierebuli eqneba namdvili arsi

- 295 - 

konkretuli Sinaarsi da upiratesad mas daemyareba da ara 

sityvierad Camoyalibebul zogad wesebs. 

vTqvaT, mocemulia amocana: `pirveli Sesakrebi tolia 3-isa, 

xolo jami tolia 9-isa. risi tolia meore Sesakrebi?~ 

mosalodneli amoxsnaa: _ 3-s 6 unda mivumatoT, raTa miviRoT 

9. Tuki romelime moswavle am droisaTvis Tavad mixvdeba, rom 

saWiro iyo gamokleba (`9-s gamovakeli 3 da ase gavige~), masw.-i 

mas Seaqebs (Tuki darwmunebulia, rom moswavles vinmem ar aswavla 

es wesi meqanikurad!) oRond, sxva moswavleebs ar mosTxovs 

maincadamainc ase amoxsnas. bavSvisaTvis amoxsnis bunebrivi gza ki 

aseTia: romelia is ricxvi, romelic 3-s unda mivumatoT, raTa 

9 miviRoT? es ricxvia 6. swored am gzas uCvenebs moswavles 

Cveneuli mTw da mTeli meTodika. 

maSasadame, ucnobi Sesakrebis moZebnis amocanas moswavleTa 

umravle-soba albaT SekrebiT amoxsnis da ara gamoklebiT! es 

bunebrivia da kargia. amdagvarive viTarebaa ucnobi maklebisa Tu 

saklebis moZebnis amocanebSic romlebsac moswavleTa 

umravlesoba amoxsnis, albaT, gamoklebiT, da ara SekrebiT.. 

ucnobi komponentis moZebnis amocanebs ukavSirdeba TamaSebi 

`Cavifiqre ricxvi~. roca klass vaxsnevinebT amocanas ucnobi Sesakrebis 

moZebnaze (mocemuli jamiTa da erTerTi SesakrebiT) sinamdvileSi 

vaxsnevinebT amocanas gamoklebaze. magram amis xazgasma 

araa saWiro. rogorc iTqva, moswavleebis umravlesoba albaT amoxsnis 

ara gamoklebiT, aramed `jamamde SevsebiT~ ramdeni aklia jamamde, 

ramdeni unda mivumatoT, raTa jami miviRoT. amaSi daexmarebaT 

mTw-s TvalsaCinoebac. magram TandaTan es amocanebi imgvari 

xdeba, rom isini uSualod gadaizrdeba gamoklebis amocanebSi. 

moswavle bunebrivad swavlobs imas, rom ucnobi Sesakrebis 

samoZebnelad gamoklebaa saWiro. pirdapir, mzamzareulad, amis 

swavleba araa saWiro (iseve rogorc sxva sakiTxebisa!). 

Cveneuli meTodikisaTvis es xom zogadi wesia: masw.-i klass ar 

awvdis gamzadebul wesebs, moswavleebi TviTon eZieben da

- 296 - 

pouloben maT. sul bolos wesis sityvierad Camoyalibebasac 

Seecdebian, oRond mravali moswavlisaTvis es SeiZleba moxdes 

meore an sulac mesame klasSi roca misi goneba bunebrivad 

momwifdeba saamisod. 

amocanis amoxsnis (ufro zustad, pasuxis moZebnis) Semdeg masw.-i 

dasvams SekiTxvas `rogor gaige?~ es SekiTxva sinamdvileSi 

gulisxmobs pasuxis mxolod Semowmebas! moswavleTa umravlesoba 

ver gaiazrebs, rom ucnobi Sesakrebis samoZebnelad gamoklebaa 

saWiro, xolo ucnobi saklebis samoZebnelad Sekreba. es jer 

arcaa aucilebeli. erTaderTi, masw.-i damatebiT ikiTxavs _ vinmes 

sxvagvarad xom ar amouxsnia? {8-s gamoklebuli 3 tolia 5isa} 

Zalian kargia! 

daaxloebiT aseve iqneba ucnobi saklebis moZebnis amocanebis 

SemTxvevaSic. xolo ucnobi maklebis moZebnis amocanebi Zalian 

advilebia. TviT maTi sayofacxovrebo Sinaarsi hkarnaxobs 

moswavles, Tu ra unda gaakeTos. arsebiTad, amgvari amocanebi 

klass ukve amoxsnilic ki aqvs. 

sazogadod, sxva sakiTxebTan dakavSirebiTac asea: moswavlis 

cnobierebaze Zaladoba araa saWiro. amoxsnas ise, rogorc bunebrivad 

eadvileba. mTavaria, moqmedebdes gaazrebulad da gabedulad. 

mTavaria, azrovnebdes. 

------------------------------------------------------------------------------------ 

229/3/!fsUfsUj!vdopcj!Tftblsfcjt!np[fcob!ojwUfc{f!nprnfef. 

cjtbt/! ! _ gamovides nukri da da magidaze dados sami kubi. 

kargi, axla me davumateb ramdenime kubs (daamatebs 5 kubs ise, 

rom bavSvebma ver daTvalon). vin gamoicnobs, ramdeni kubi 

damidia me? magram jer gavixsenoT: ramdeni kubi dado nukrim? 

{sami}. maS me ramdeni damidia? 

Tuki vercerTi moswavle ver mixvdeba, masw.-i Tavad 

ikiTxavs: _ axla ramdenia? {rva} ese igi, me ramdeni damidia? 

vin gamoicnobs? (Tuki gauWirdebaT, daixmarion mTw-Si `8~). 

Sesruldeba kidev erTi amgvari savarjiSo, raodenoba 3 

Seicvleba 2-iT, 5 Seicvleba 7-iT. oRond, Sesrulebisas Tan

- 297 - 

Seivseba saTanado sqema: dafaze daixazeba erTi imdagvari sqema, 

rogorebicaa msr-Si (118.2). _ ramdeni kubi dado nukrim? 

{ori} hoda CavweroT aq (2-ians Cawers sul marcxena CarCoSi, 

moswavleebi Caweren msr-s sqemaSi, iqac sul marcxena 

CarCoSi). me farulad davde ramdeniRac kubi, Tqven ar 

daginaxavT (da meore CarCoSi Caiwereba ? niSani). ratom 

Cavwere aq es kiTxvis niSani? {imitom, rom moswavleebma ar 

viciT, masw.-ma ramdeni kubi dado} mere, bolos, ramdeni kubi 

aRmoCnda magidaze? {cxra} hoda am jamSi CavweroT 9-iani 

{Caiwereba qvemoT elifsSi _ dafazec da msr-Sic} amocana ras 

gvekiTxeba? {ramdeni kubi daudia masw.-s _ Tan TiTs daadebs ? 

niSans} Teklem gaimeoros mTliani amocana! kargi, vin amoxsnis? 

(mTw gadaSalon `9~-ze) {masw.-s daudia Svidi kubi} saidan 

iciT? {ors mivumatoT Svidi tolia cxrisa} diax, CavweroT aq 

7-iani {waSlis ? niSans da dawers 7-s, moswavleebi msr-Si ? 

niSans gadaaweren 7-ians}. 

_ axla me gikarnaxebT erT aseT amocanas da Tqven Sin unda 

amoxsnaT (118.9). moswavlem magidaze dado 4 kubi. CawereT 

CarCoSi 4-iani! sad unda CaweroT? diax. masw.-ma farulad 

daumata ramdeniRac kubi. sad ra unda CaweroT? {meore 

CarCoSi CavweroT ? niSani} diax! amis Semdeg magidaze aRmoCnda 

7 kubi. sad unda CaweroT 7-iani? ras gvekiTxeba amocana? 

{ramdeni kubi daudia masw.-s} diax, amas Sin gamoicnobT da 

CawerT kiTxvis niSnis adgilas, zed gadaawerT. 

229/4/! npoblwfUjt! hb{pnwb! eb! npdfnvmj! tjhs[jt! npoblwfUjt! 

ebyb{wb-! :! tn.jt! gbshmfcTj/ klasi Cveulebrivad swavlobs 

patara saxazaviT monakveTis gazomvas _ gazomaven msr-Si daxazulebs 

(118.3). masw.-i, rogorc yovelTvis, merxebs Soris 

Camoivlis da Seamowmebs, kargad zomaven Tu ara. Semdeg iswavlian 

mocemuli sigrZis monakveTis daxazvas. masw.-i 

ukarnaxebs sigrZes da moswavleebi 118.3-is cariel adgilas 

daxazaven: _ 2 sm; 4 sm; 1 sm; 9 sm (es grZeli monakveTi 

daxazon rveulis furclis zeda an qveda mindorze) da sxva. 

229/5/!`txsbgj!qbtvyfcj~!`!bohbsjTj/ SekiTxvebia Cveulebrivi

- 298 - 

saangriSo: sams mivumatoT rva; aTs gamovakloT oTxi da ase 

Semdeg _ cxrametis farglebSi.! 

g 119. `psj!bUfvmj!eb!ovmj~-!npd/!hwfsejt!tbnlvUy/!ebyb{wb/ 

22:/2/! pdjbo! hspwbTj! bUfvmjt! hbnpzpgb/! kvgUfcjU! ebUwmb;! 

`psj! dbmj! bUfvmj! eb! obTUj! ovmj~/! nplmfe;! `psj! dbmj! 

bUfvmj!eb!ovmj!~-!bov;!`bUj!eb!bUj~/!`sbnefojU!nfujb!bU{f@~! 

`!`bUjU!nfuj~/!pdjt!Tfebsfcb!obtxbwm!sjdywfcUbo/!!yvelaferi 

gakeTdeba iseve, rogorc 92.1-Si, oRond ocisaTvis. 

22:/3/! fsUfsUj! vdopcj! Tftblsfcjt! np[fcob! tbzpgbdypwsfcp! 

bnpdbofcTj/! jer Sesruldeba erTi imdagvarive savarjiSo, 

rogoric iyo 118.2-Si (kubebiT), magaliTad, 3 + ? = 9. sqemis 

daxazva araa saWiro, daixmarion mTw. 

Semdeg dafaze daixazeba erTi imdagvari sqema, rogorebicaa 

msr-Si (119.2). _ or Zmas erTad hyavT 8 Svili (da masw.-i 

sqemis qveda `jamSi~ anu elifsSi Cawers 8-ians). ufros Zmas 

hyavs 3 Svili (da masw.-i sqemis marcxena CarCoSi Cawers 3ians). 

ramdeni Svili hyolia umcros Zmas? sofiko, ucqire am 

naxazs da srulad waikiTxe amocanis piroba! kargi, axla 

amovxsnaT amocana. (amoixsneba da cariel CarCoSi Caiwereba 

pasuxi _ 5). rogor gaige? {albaT ase: 3-s 5 unda mivumatoT, 

raTa miviRoT 8}. 

SekiTxva `rogor gaige~ sinamdvileSi gulisxmobs pasuxis 

Semowmebas! moswavleTa umravlesoba ver gaiazrebs, rom aq 

gamoklebaa saWiro. es jer arcaa aucilebeli. _ vinmes 

sxvagvarad xom ar amouxsnia? {8-s gamoklebuli 3 tolia 5isa} 

Zalian kargia, yoCaR! 

Semdeg amocanas moswavleebi msr-s pirvel, marcxena 

sqemaSi (119.2) Seitanen da ise amoxsnian: 

2) Wrelma dedalma sami dRis ganmavlobaSi TiTo kvercxi 

dado. sul ramdeni kvercxi daudia mas? {sami} CavweroT es 3iani 

am CarCoSi! mis garda kvercxebs Savi dedalic debda. sul 

ki sabudarSi 7 kvercxi iyo. sad unda Caiweros 7-iani? {`jam- 

Si~} CavweroT! ramdeni kvercxi daudia Sav dedals? konstan-

- 299 - 

tine, gaimeore amocana! Sexede am naxazs da ise mipasuxe! 

Semdeg amocanas ki mxolod meore sqemaSi Seitanen, xolo 

amoxsniT _ saSinao davalebaSi amoxsnian. 

3) da da Zma erTad wvetdnen nesvebs, sul 8 nesvi moitanes. 

Zmam daTvala, rom mas 4 nesvi mouwyvetia. ramdeni nesvi 

mouwyvetia das? 

aq erTi mcire cvlilebaa: ? niSani Caiwereba ara marjvena, 

aramed marcxena CarCoSi. es imitom, rom amocanis pirobaSi 

jer daa naxsenebi, Semdeg ki _ Zma. ese igi, ucnobia pirveli 

Sesakrebi. 

22:/4/!npdfnvmj!tjhs[jt!hwfsejt!nrpof!tbnlvUyfejt!ebyb{wb-!!!!! 

:! tn.jt! gbshmfcTj/! moswavleebi 119.3-is cariel adgilas 

daxazaven da gaaferadeben samkuTxedebs, romelTa cali 

gverdis sigrZes masw.-i ukarnaxebs: 2 sm, 4 sm, 3 sm, 2 sm. 

danarCeni gverdebi daxazon ise, rogorc gamouvaT. 

22:/5/! `txsbgj! qbtvyfcj! `! lblvo.ubTfcj~/ ix. 18.4, oRond 

raodenoba aRwevs ocamde. 

g 120.!{fq/!Tfls/!npd/!psj!hwfsejU!pUylvUyfejt!ebyb{wb/ 

231/2/! {fqjsj! Tflsfcb: 10+10, 19+1, 1+19.! am sagakveTilo 

monakveTs cota dro eyofa. gavameorebinebT kvlav oceuls 

(rogorcaa 119.1-Si). Semdeg masw.-i dafaze daxatavs patara 

wreebis `dominosebur~ oceuls. _ SexedeT, ramdeni wrea aq? 

{oci} ramdenad da ramdenadaa dalagebuli oceuli? (masw.-i Tan 

TiTiT cal-calke Semoxazavs aTeulTa CarCoebs) {aTeulad da 

aTeulad} ese igi, ris tolia aTs mimatebuli aTi? {ocisa} 

gaimeoros leqsom! kargi, axla moifiqreT, ricxvTa rigSi 

romel ricxvs mosdevs 20? {cxramets} ese igi, ris toli 

yofila 19+1? {ocisa} gaimeoros natom! 1+19 risi toli 

iqneba? {ocisa} saidan ici? {1+19 tolia 19+1-isa} axla 

Sekrebis samive es magaliTi erTad gaimeoros SoTam! 

231/3/!fsUfsUj!vdopcj! Tftblsfcjt! np[fcob/ mTw gadaiSleba 

`11”-ze. masw.-i rig-rigobiT ekiTxeba sxvadasxva moswavleebs 

(moswavleebi sxedan da mTw-s akvirdebian, pasuxoben

- 300 - 

mjdomare) _ ramdeni aklia 8-s 11-mde? ramdeni unda mivumatoT 

8-s, raTa 11 mi-viRoT? ramdeni aklia 7-s 11-mde? ramdeniT 

unda gavadidoT 7, raTa 11 miviRoT? pirveli Sesakrebi tolia 

7-isa, jami ki tolia 11-isa. risi tolia meore Sesakrebi? 

ramdeni aklia 9-s 13-mde? ramdeniT unda gavadidoT cxra, 

raTa 13 miviRoT? pirveli Sesakrebi tolia 9-isa, jami ki 

tolia 13-isa. risi tolia meore Sesakrebi? ramdeni aklia 5-s 

12-mde? ramdeni unda mivumatoT 5-s, raTa 12 miviRoT? 

Tuki moswavles gauWirdeba pasuxis gacema, mTw 

gadaiSleba saTanado gverdze _ ramdenicaa jami. Tuki 

romelime moswavles mainc gauWireba pasuxis gacema, masw.-i 

hkiTxavs: _ aba am gverdze moZebne cxaeuli! diax, esaa. sul 

jamSi ramdeni wrea? {13} ese igi, ramdeni aklia 9-s 13-mde? 

{4} ramdeni unda mivumatoT cxras, raTa 13 miviRoT? {4} 

kargi, axla dafiqrdi: pirveli Sesakrebi tolia 7-isa, jami ki 

tolia 11-isa. risi tolia meore Sesakrebi? maS, ramdeniT unda 

gaadido cxra, raTa 13 miiRo? 

Semdeg SekiTxvebi cotaTi icvleba: _ ras unda mivumatoT 

8, raTa 14 miviRoT? meore Sesakrebia 8, jamia 14, risi tolia 

pirveli Sesakrebi? ras unda mivumatoT 6, raTa 16 miviRoT? 

meore Sesakrebia 6, jamia 16, risi tolia pirveli Sesakrebi? 

ras unda mivumatoT 9, raTa 17 miviRoT? meore Sesakrebia 9, 

jamia 17, risi tolia pirveli Sesakrebi? ras unda mivumatoT 

7, raTa 15 miviRoT? meore Sesakrebia 7, jamia 15, risi tolia 

pirveli Sesakrebi? (es yvelaferi zepirad keTdeba. 

amitom araa saWiro ricxvebis Cawera!). 

bolos, masw.-i dafaze daxazavs erT imdagvar sqemas, 

rogoracaa msr-Si erTi sqema (120.2). _ me Cavifiqre raRac 

ricxvi (da TiTs daadebs sqemaSi mkrTal ?-s). mivumate xuTi 

(Cawers CarCoSi 5-ians) da jamSi miviRe 9 (`jamSi~ anu elifs- 

Si Cawers 9-ians). vin gamoicnobs _ ra ricxvi Camifiqrebia? 

(daadebs TiTs ?-s). ra ricxvi unda eweros aq? (roca moswavle 

upasuxebs, mkrTal ?-s zemodan gadaawers 4-ians). rogor 

gaige? {albaT ase: 5-s 4 unda mivumatoT, raTa miviRoT 9}.

- 301 - 

rogorc iTqva zemore, II-13-Si, amgvari dasabuTebisa da 

gamoTvlis saSualeba moswavles ar unda wavarTvaT. miT 

umetes, rom mTw mas am gzaze ayenebs. Sesworeba araa saWiro. 

xolo SekiTxva `rogor gaige?~ sinamdvileSi gulisxmobs 

pasuxis mxolod Semowmebas! 

Semdeg masw.-i klass ukarnaxebs erT amgvarsave amocanas 

(ricxvis Cafiqrebaze), mis monacemebs Seitanen msr-s meore, 

marjvena sqemaSi, xolo amoxsniT _ saSinao davalebaSi 

amoxsnian (120.9): 

_ me Cavifiqre ricxvi 4. CawereT CarCoSi 4-iani! sad unda 

CaweroT? diax. mivumate raRac ricxvi. ? niSani ukve weria, 

Tqven nuRar dawerT. mivumate da miviRe jamSi 8. CawereT 

jamSi 8-iani! Sin ki gamoicnobT, ra ricxvi unda eweros ? 

niSnis nacvlad! 

231/4/! npdfnvmj! tjhs[jt! psj! hwfsejt! nrpof! pUylvUyfejt! 

ebyb{wb-!:!tn.jt!gbshmfcTj/ moswavleebi 120.3-is cariel adgilas 

daxazaven da gaaferadeben oTxkuTxedebs, romelTa ori 

gverdis sigrZes masw.-i ukarnaxebs: 1 sm da 3 sm; 2 sm da 2 sm; 1sm 

da 2 sm. danarCeni gverdebi daxazon ise, rogorc gamouvaT. 

231/5/! `txsbgj! qbtvyfcj! `! sjdywjt! eblblvofcb~/ ix. 102.4, 

oRond ocis CaTvliT. 

g 121. {fqjsj!hbnplmfcb/!vdopcj!nblmfcjt!np[fcob/! 

232/2/! {fqjsj! hbnplmfcb;! 31`21-! 31`2/! am sagakveTilo 

monakveTs cota dro eyofa. jer swrafad gavameorebinebT 120.1s, 

Semdeg moswavleebi advilad gamoTvlian mocemul or 

sxvaobas. 

moswavle. 

TviTeuls srulad gaimeorebs romelime sxva 

232/3/! vdopcj! nblmfcjt! np[fcob! ojwUfc{f! nprnfefcjtbt/! _ 

gamovides nunu da aq dados aTi kubi. kargi, axla me aviReb 

ramdenime kubs (aiRebs 7 kubs ise, rom bavSvebma ver 

daTvalon). vin gamoicnobs, ramdeni kubi amiRia me? jer 

gavixsenoT, ramdeni kubi dado nunum? {aTi}. 

Tuki vercerTi moswavle ver mixvdeba, masw.-i Tavad ikiTxavs: _

- 302 - 

axla ramdeni darCa? {sami} ese igi, me ramdeni amiRia? vin 

gamoicnobs? (Tuki gauWirdebaT, daixmarion mTw-Si `10~). 

Sesruldeba kidev erTi amgvari savarjiSo, raodenoba 10 

Seicvleba 12-iT, 7 Seicvleba 4-iT. oRond, Sesrulebisas Tan 

Seivseba saTanado sqema: dafaze daixazeba erTi imdagvari sqema, 

rogorebicaa msr-Si (121.2). _ ramdeni kubi dado nukrim? 

{Tormeti} hoda CavweroT aq (12-s Cawers sul marcxena 

`jamSi~, moswavleebi Caweren msr-s sqemaSi, iqac sul marcxena 

`jamSi~). ratom weria aq Semdeg gamoklebis niSani da kiTxvis 

niSani? {imitom, rom masw.-ma farulad aiRo ramdeniRac kubi, 

moswavleebs ar dagvinaxavs da unda gamovicnoT} mere, bolos, 

ramdeni kubi aRmoCnda magidaze? {rva} hoda aq CavweroT 8-iani 

{Caiwereba gverdiT CarCoSi _ dafazec da msr-Sic} amocana 

ras gvekiTxeba? {ramdeni kubi daudia masw.-s _ Tan TiTs 

daadebs ? niSans} luka, ucqire am naxazs da srulad waikiTxe 

amocanis piroba! kargi, axla amovxsnaT amocana. kargi, vin 

amoxsnis? (mTw gadaSalon `8~-ze) {masw.-s auRia oTxi kubi} 

saidan iciT? {Tormets gamovakloT oTxi tolia rvisa} diax, 

CavweroT aq 4-iani {waSlis ? niSans da dawers 4-s, 

moswavleebi msr-Si ? niSans gadaaweren 4-ians}. 

_ axla me gikarnaxebT erT aseT amocanas da Tqven Sin unda 

amoxsnaT. moswavlem magidaze dado 11 kubi. CawereT `jamSi~ 11! 

sad unda CaweroT? diax. masw.-ma farulad aiRo ramdeniRac 

kubi. amitomac weria aq gamoklebis niSani da ? niSani. amis 

Semdeg magidaze aRmoCnda 7 kubi. sad unda CaweroT 7-iani? 

ras gvekiTxeba amocana? {ramdeni kubi auRia masw.-s} diax, amas 

Sin gamoicnobT da CawerT kiTxvis niSnis adgilas, zed 

gadaawerT. 

232/4/!npdfnvmj!tjhs[jt!npoblwfUUb!nrpof!hbytojmj!ufyjmf. 

cjt! ebyb{wb-! :! tn.jt! gbshmfcTj/ moswavleebi 121.3-is cariel 

adgilas daxazaven da gaaferadeben gaxsnil texilebs, romelTa 

Semadgeneli monakveTebis sigrZeebs masw.-i ukarnaxebs: 

1 sm, 2 sm, amdenive, 1 sm; 3 sm, 1 sm, 2 sm, amdenive. 

232/5/! `txsbgj! qbtvyfcj! `! Dbwjgjrsf! sjdywj~/ _ Cavifiqre

- 303 - 

ricxvi, mivumate 6, miviRe 11. ra ricxvi Camifiqrebia? (da ase 

Semdeg _ naswavl magaliTebze). pirvel wreze advilebia: 

mivumate 2, miviRe 8. Semdeg TandaTan rTuldeba. 

g 122. U/!23/2.jt!hbnfpsfcb/!npd/!hwfsejt!lwbesbujt!ebyb{wb/ 

233/1/ 121.8 kargad unda gairCes. _ axla kidev erTxel 

daakvirdiT Sevsebul cxrils. moZebneT cifri 3. mis qveS ra 

raodenobebia? {sameuli da cameteuli} ratomaa swored eseni 

3-ianis svetSi? {imitom, rom sameuli 3-is toli raodenobaa, 

xolo camets ki sami naSTad aqvs}. 

aseve ganixileba kidev ramdenime sveti. cxadia, sxvadasxva 

moswavleebi upasuxeben. 

233/2/!Ubwj!23/2.jt!Tfkbnfcb!eb!hbnfpsfcb/!!!davaTvalierebinebT 

mTw-s `11~-dan `19-20~-mde. TiTo ricxvze cota xans 

SevCerdebiT: swrafad vaangariSebinebT or-sam Sekrebagamoklebas. 

sanam axal ricxvze gavavalT, masw.-i ikiTxavs: _ 

maS ratom hqvia am ricxvs `T-erT-meti~? 

bolos daemateba amgvari saazrovno SekiTxva: _ ramdeni 

TiTi gvaqvs orive xelze da orive fexze erTad? saidan ici? 

{xelexze gvaqvs aTi TiTi, fexebzec _ amdenive, sul ocia}. 

233/3/!vdopcj!nblmfcjt!np[fcob!tbzpgbdypwsfcp!bnpdbofcTj/ 

jer Sesruldeba erTi imdagvarive savarjiSo, rogoric iyo 

121.2-Si (kubebiT), magaliTad, 13 _ ? = 5. sqemis daxazva araa 

saWiro, daixmarion mTw. 

Semdeg masw.-i dafaze daxazavs erT imdagvar sqemas, rogoricaa 

msr-Si marcxena sqema (122.2). masw.-i am sqemas Seavsebs, moswavleebi 

_ msr-s pirveli, marcxena sqemas (`naxazs~). 

1) ezoSi 9 manqana idga (masw.-i 9-ians marcxena `jamSi~ 

Cawers, moswavleebi _ msr-s marcxena `jamSi~). cota xnis 

Semdeg ezoSi sami manqana darCa (CarCoSi Caiwereba 3-iani). 

ramdeni manqana wasula ezodan (masw.-i TiTs daadebs mkrTal 

?-s)? iraklim Sexedos am naxazs da gaimeoros amocanis

- 304 - 

piroba! (Semdeg amocanas amoxsnian da pasuxi 6 zedve 

gadaewereba mkrTal ?-s). 

Semdeg amocanas ki mxolod meore, marjvena sqemaSi Seitanen, 

xolo amoxsniT _ saSinao davalebaSi amoxsnian (122.9): 

2) bicolam sinze 8 naWeri namcxvari dado. roca iq 

bavSvebi mividnen, sinze mxolod 5 namcxvariRa daxvdaT. 

bavSvebamde oTaxSi fiso Separula! ramdeni namcxvari 

mouparavs fisos? 

233/4/! npdfnvmj! tjhs[jt! hwfsejt! nrpof! lwbesbujt! ebyb{wb! 

vsfefcjbo! gvsdfm{f/! moswavleebi 122.3-is cariel badeSi 

daxazaven da gaaferadeben kvadratebs, romelTa gverdis 

sigrZeebs masw.-i ukarnaxebs: 

1 sm, 2 sm, amdenive, 3 sm. 

233/5/! `txsbgj! qbtvyfcj! `! sjdywfcjt! ubTfcj/~ am TamaSSi rigrigobiT 

erTi TiTo moswavle ki ar pasuxobs, aramed _ ori-ori 

(erT merxze msxdarni). masw.-i daasaxelebs raime ricxvs 

ocdacxramde (amjerad _ mxolod ocis CaTvliT _ sadamdec 

naswavli aqvT). moswavleTa wyvilidan marcxenam unda Semokras 

taSi imdenjer, ramdeni aTeulicaa am ricxvSi, xolo marjvenam _ 

am aTeulTa garda naSTi risi tolicaa. magaliTebi: 

_ TxuTmeti {taSi erTxel, taSi xuTjer}; eqvsi {siCume, taSi 

eqvsjer}; ocdaori {taSi orjer, taSi orjer} aTi {taSi erTxel, 

siCume}; nuli {siCume, siCume}; ToTxmeti {taSi erTxel, 

taSi oTxjer}; oci {taSi orjer, siCume}; da ase Semdeg. 

es TamaSi saguldagulod unda Catardes, radganac 123.1-is 

Semamzadebeli safexuria! 

------------------------------------------------------------------------------------- 

I-13. 21.ebo!31.nef!sjdywfcjt!djgsvmj!Dbxfsb

- 305 - 

kargad gaiazros ricxvis aTobiTi cifruli Canaweri (mxolod 20is 

farglebSi es uaresad Cans); da raTa Semzaddes meore klasis 

ariTmetikis xazi moswavle igrZnobs, rom ricxvebi aseve 

grZeldeba. amasTan, 30-mde ricxvebi ar Seicavs araviTar damatebiT 

siZneles, 20-mde ricxvebTan SedarebiT. 21-dan 29-mde ricxvebi ki 

iswavleba, magram SemdgomSi maT iSviaTad vaxsenebT ZiriTadad 

mainc 20-is farglebSi davrCebiT. 

jer vaswavliT cifrul Canawerebs 12, 13, ..., 19 (xolo 10, 11 da 

20 cota ufro rTulia da momdevno gakveTilze iqneba). saWiroa 

imis gaazreba, rom marcxena cifri 1 gviCvenebs, Tu ricxvSi 

ramdenia aTeuli; xolo marjvena cifri (magaliTad, 2) gviCvenebs, 

Tu ra rCeba naSTad. ese igi, ricxvSi 12 (swored ricxvSi, 

raodenobaSi da ara mis cifrul CanawerSi!) aris 1 aTeuli da 

kidev naSTi 2. 

Zalian gavrcelebuli Secdomaa, roca amboben: ricxvSi 12 aris 

1 aTeuli da 2 erTeuli. anda: cifri 1-iani gviCvenebs aTeulTa 

raodenobas, xolo 2-iani erTeulTa raodenobas. es uxeSi 

Secdomaa. sinamdvileSi ricxvSi 12 aris 12 erTeuli! 2 aris ara 

erTeulTa raodenoba, aramed aTeulebs morCenil erTeulTa 

raodenoba. swored esaa naSTi. xolo termini èrTeuli~ 

ariTmetikuli azriT pirvel klasSi sruliad zedmetia. miT 

umetes, rom Cven gvaqvs sigrZis erTeulebi dm da sm! bavSvi veRar 

gaigebs, romelia es èrTeuli~ da ratom ewodeba mas ase! 

amrigad: 1 aTeuli da naSTi 2. 

------------------------------------------------------------------------------------ 

g 123. vdopcj!nblmfcjt!np[fcob/!npd/!hwfse/!nbsUlvU/!ebyb{wb/ 

234/2/!djgsvmj!Dbobxfsfcj;!23-!24-!25-!26-!27-!28-!29-!2:/!nbUj! 

hbb{sfcb/! _ romeli ricxvebis cifruli Cawera gvaqvs ukve 

naswavli? {0, 1, ... , 9} romlebia is ricxvebi, romlebic 

naswavli ki gvaqvs, magram maTi cifruli Cawera rom ar 

gviswavlia? {10, 11, ... , 20} dRes viswavliT cifrebiT Caweras 

Tormetisa, cametisa, ToTxmetisa da ase Semdeg cxrametamde.

- 306 - 

aba, vin icis, rogor iwereba cifrebiT 12? 

SeuZlebelia, rom es romelime moswavlem mainc ar icodes, 

Tundac mTw-s Tvalierebis gamo. mTw gadaiSalos `12~-ze. 

TiTiT aCvenon, romelia Tormetis cifruli Canaweri. masw.-i ki 

msxvilad dawers dafaze. _ Tu iciT, es 1 ras gviCvenebs? aba 

daxedeT, Tormeteuli risgan da risgan Sedgeba? {aTeulisa da 

naSTi orisagan} diax, ramdeni aTeulia TormetSi? {erTi} 

naSTi? {ori} ese igi, es 1-iani ras gviCvenebs? {TormetSi 

ramdeni aTeulicaa} es 2-iani? {aTeulTa garda rac rCeba 

naSTad} kargi, axla moZebneT cametis cifruli Canaweri! diax, 

esaa, mec davwer dafaze. ras gviCvenebs es 1-iani? {cametSi 

ramdeni aTeulicaa} es 3-iani? {aTeulTa garda rac rCeba 

naSTad} kargi, axla moZebneT ToTxmetis cifruli Canaweri! 

(da ase Semdeg cxrametis CaTvliT _ Semdgomi ricxvebi ukve 

swrafad wava). 

Semdeg gadaSlian msr-s (123.1) da TviTeuli `dominos~ 

qveS daweren Sesabamis ricxvs cifrebiT. 

234/3/! vdopcj! nblmfcjt! np[fcob/! mTw gadaiSleba `12~-ze. 

masw.-i rig-rigobiT ekiTxeba sxvadasxva moswavleebs 

(moswavleebi sxedan da mTw-s akvirdebian, pasuxoben 

mjdomare) _ ramdeniT metia 12, vidre 8? ramdeni unda 

gamovakloT 12-s, raTa 8 miviRoT? ramdeniT metia 12, vidre 

5? ramdeniT unda SevamciroT 12, raTa 5 miviRoT? saklebia 13, 

sxvaobaa 7. risi tolia maklebi? ramdeniT unda SevamciroT 13, 

raTa 7 miviRoT? 

Tuki moswavles gauWirdeba pasuxis gacema, mTw 

gadaiSleba saTanado gverdze _ ramdenicaa jami. Tuki 

romelime moswavles mainc gauWireba pasuxis gacema, masw.-i 

hkiTxavs: _ aba am gverdze moZebne Svideuli! diax, esaa. sul 

jamSi ramdeni wrea? {13} ese igi, ramdeni unda gamovakloT 13s, 

raTa 7 miviRoT? {6} kargi, axla dafiqrdi: saklebia 13, 

sxvaobaa 7. risi tolia maklebi? maS ramdeniT unda 

SevamciroT 13, raTa 7 miviRoT?

- 307 - 

es yvelaferi zepirad keTdeba. ricxvebis Cawera araa 

saWiro! 


rogoricaa msr-Si marcxena sqema (123.2). moswavleebi ki msr- 

Sive Seavseben. _ me Cavifiqre ricxvi 15 (marcxena `jamSi~ 

Caiwereba 15), gamovakeli raRac ricxvi (da TiTs daadebs 

sqemaSi mkrTal ?-s). da sxvaoba miviRe 9 (CarCoSi Caiwereba 

9). vin gamoicnobs _ ra ricxvi gamomiklia? (daadebs TiTs ?s). 

ra ricxvi unda eweros aq? (roca moswavle upasuxebs, 

mkrTal ?-s zemodan gadaawers 6-ians). rogor gaige? {albaT 

ase: 15-s 6 unda gamovakloT, raTa miviRoT 9}. SoTa, ucqire 

am naxazs da srulad waikiTxe amocanis pirobac da pasuxic! 





pasuxis mxolod Semowmebas! 

Semdeg masw.-i klass ukarnaxebs erT amgvarsave amocanas (raRac 

ricxvis gamoklebaze), maT monacemebs Seitanen msr-s meore 

sqemaSi, xolo amoxsniT _ saSinao davalebaSi amoxsnian (123.9). 

234/4/! npdfnvmj! tjhs[jt! hwfsefcjt! nrpof! nbsUlvUyfejt! 

ebyb{wb!vksfefcjbo!gvsdfm{f/!moswavleebi 123.3-is cariel badeSi 

daxazaven da gaaferadeben marTkuTxedebs, romelTa gverdis 

sigrZeebs masw.-i ukarnaxebs: 2 sm da 3 sm; 4 sm da 1 sm. 

234/5/!`txsbgj!qbtvyfcj!`!Dbwjgjrsf!sjdywj~/!!!! _ Cavifiqre 

ricxvi, mivumate 6, miviRe 11. ra ricxvi Camifiqrebia? (da ase 

Semdeg _ naswavl magaliTebze). 

234/9/ kanonzomierebaa: marTkuTxedi, Sekruli texili, 

gaxsnili mrudi, da ase Semdeg. 

g 124. vdopcj!tblmfcjt!np[fcob/!ufyjmjt!tjhs[jt!hbnpUwmb/ 

235/2/! djgsvmj! Dbobxfsfcj;! 21-! 22! eb! 31/! nbUj! hbb{sfcb/! ! _ 

romeli ricxvebis cifruli Cawera viswaleT wina gakveTilze? 

{12, 13, ..., 19} romlebia is ricxvebi, romlebic naswavli ki 

gvaqvs, magram maTi cifruli Cawera jer ar gviswavlia? {10,

- 308 - 

11, 20} dRes viswavliT swored am ricxvebis cifrebiT 

Caweras! aba, vin icis, rogor iwereba cifrebiT aTi? 

mTw gadaiSalos `10~-ze. TiTiT aCvenon, romelia aTis cif 

ruli Canaweri. masw.-i ki msxvilad dawers dafaze. _ Tu iciT, 

es 1 ras gviCvenebs? {aTSi ramdeni aTeulicaa} diax, ramdeni 

aTeulia aTSi? {erTi} naSTi ra rCeba? {nuli} ese igi, es 1-iani 

ras gviCvenebs? {aTSi ramdeni aTeulicaa} es 0-iani? {aTeulTa 

garda rac rCeba naSTad} kargi, axla moZebneT TerTmetis 

cifruli Canaweri! diax, esaa, mec davwer dafaze. ras 

gviCvenebs es 1-iani? {TerTmetSi ramdeni aTeulicaa} es 1-iani? 

{aTeulTa garda rac rCeba naSTad} kargi, axla moZebneT ocis 

cifruli Canaweri! ras gviCvenebs es 2-iani? {ocSi ramdeni 

aTeulicaa} es 0-iani? {aTeulTa garda rac rCeba naSTad}. 

Tan gadaSlian msr-s (124.1) da TviTeuli `dominos~ 

gverdiT or-orjer daweren Sesabamis ricxvs cifrebiT. 

235/3/!vdopcj!tblmfcjt!np[fcob!tbzpgbdypwsfcp!bnpdbofcTj/ 

am sakiTxs aRar viwyebT nivTebze moqmedebebiT. es ukve 

sakmarisad iyo, Tanac, ucnobi saklebis moZebnis amocanis 

nivTebiT warmodgena arcaa mTlad bunebrivi. 

masw.-i dafaze daxazavs erT imdagvar sqemas, rogoracaa 

msr-Si erTerTi sqema (124.2). _ amovxsnaT amocana da Tan am 

naxazebSi CavweroT. moswavles davalebad marTkuTxedebis 

daxazva hqonda, ar viciT ramdenis (da TiTs daadebs sqemaSi 

mkrTal ?-s). man daxaza 6 marTkuTxedi (da 6-ians Cawers 

cariel CarCoSi, gamoklebis niSnis marjvniv; moswavleebic 

Caweren msr-s pirvel, marcxena sqemaSi 124.2). moswavles 

kidev darCa dasaxazavi 5 marTkuTxedi (da 5-iani Caiwereba 

rombSi). ramdeni marTkuTxedis daxazva hqonia mas davalebad? 

(da TiTs daadebs sqemaSi mkrTal ?-s). ra unda eweros aq 

kiTxvis niSnis nacvlad? axla vaxom Sexedos am naxazs da 

gaimeoros amocanis piroba! 

amoxsnis Semdeg sqemaSi mkrTal ?-s zedve gadaewereba 

pasuxi 11. _ rogor gaige pasuxi? {11-s unda gamovakloT 6, 

raTa miviRoT 5}. rogorc mravaljer iTqva, SekiTxva `rogor

- 309 - 

gaige?~ sinamdvileSi gulisxmobs pasuxis mxolod Semowmebas! 

_ vinmes sxvagvarad xom ar gaugia pasuxi? {6+5=11} yoCaR, 

Zalian kargia! 

Semdegi amocana msr-s meore, marjvena sqemaSi mxolod 

Caiwereba, xolo amoixsneba saSinao davalebaSi. 

2) papas bazarSi 5-lariani daekarga, ris Semdegac 9 lari 

darCa. ramdeni lari hqonia mas bazarSi Sesvlamde? 

235/4/!npdfnvmj!tjhs[jt!npoblwfUUb!nrpof!hbytojmj!ufyjmf. 

cjt! ebyb{wb-! spdb! tjhs[ffcj! npdfnvmjb! dysjmjt!tbyjU/! ufyj. 

mjt!nUmjboj!tjhs[jt!hbnpUwmb-!spdb!njtj!Tfnbehfofmj!zwfmb! 

npoblwfUjt!tjhs[f!tboujnfusfcjt!nUfm!sjdywt!Tfjdbwt/!moswavleebi 

124.3-is cariel adgilas daxazaven gaxsnil texils, 

romlis Semadgeneli monakveTebis sigrZeebs Tavadve amoikiTxaven 

cxrilidan 124.3. Semdeg iangariSeben mis mTlian sigrZes. 

235/5/!`txsbgj!qbtvyfcj!`!sjdywfcjt!ubTfcj/~ ix. 122.4. 

g 125. Dbobxfsfcj;!32-!33-!///!-!3:/!npoblwfU/!{pnwb!gvsdfm{f/! 

236/2/!sjdywfcj!eb!nbUj!djgsvmj!Dbobxfsfcj;!32-!33-!///!-!3:/ 

_ daxedeT kidev am lokokinas niJarasaviT naxazs (msr-s 

124.8). ocis Semdeg Tu grZeldeba ricxvebi? romlebia isini? 

ai magaliTad, ocdaerTi. daukvirdiT am sityvas: oc-da-erTi. 

vin dasvams dafaze amden wertilss? (gamova moswavle da 

`dominoseburad~ dasvams oc wertils, Semdeg gverdiT 

dausvams kidev erTs). ramdeni aTeulia am raodenobaSi? {ori 

aTeulia} naSTi risi tolia? {erTisa} ese igi, rogor unda 

CavweroT cifrebiT es raodenoba? {21} es 2-iani ras 

gviCvenebs? {21-Si aTeulTa raodenobas} es 1-iani ras 

gviCvenebs? {ramdenicaa naSTi}. axla vin dasvams ocdaxuT 

wertils? ramdeni aTeulia am raodenobaSi? {ori aTeulia} 

naSTi risi tolia? {xuTisa} ese igi, rogor unda CavweroT 

cifrebiT es raodenoba? {25} es 2-iani ras gviCvenebs? {25-Si 

aTeulTa raodenobas} es 5-iani ras gviCvenebs? {ramdenicaa 

naSTi}. vin metyvis, rogor iwereba ocdaori? {2 da 2}

- 310 - 

ocdaSvidi? {2 da 7}. 

Semdeg msr-Si (125.1) TviTeuli `dominos~ qveS daweren 

Sesabamis ricxvs cifrebiT. 

236/3/! vdopcj! tblmfcjt! np[fcob/! mTw gadaiSleba `15~-ze. 

masw.-i rig-rigobiT ekiTxeba sxvadasxva moswavleebs (moswavleebi 

sxedan da mTw-s akvirdebian, pasuxoben mjdomare) _ 

ras unda gamovakloT 6, raTa 9 miviRoT? romeli ricxvi unda 

SevamciroT 6-iT, raTa 9 miviRoT? maklebia 7, sxvaobaa 8. 

risi tolia saklebi? romeli ricxvi unda SevamciroT 7-iT, 

raTa 8 miviRoT? 

Tuki romelime moswavles gauWirdeba pasuxis gacema, 

maswav-lebeli hkiTxavs: _ aba am gverdze moZebne Svideuli da 

rvaeuli! diax, esaa. sul jamSi ramdeni wrea? {15} ese igi, ras 

unda gamovakloT 7, raTa 8 miviRoT? romeli ricxvi unda 

SevamciroT 7-iT, raTa 8 miviRoT? kargi, axla dafiqrdi: 

maklebia 7, sxvaobaa 8. risi tolia saklebi? 

es yvelaferi zepirad keTdeba. ricxvebis Cawera araa saWiro! 


rogoracaa msr-Si marcxena sqema (125.2). moswavleebi ki msr- 

Si Seavseben `naxazs~. _ me Cavifiqre raRac ricxvi (da TiTs 

daadebs sqemaSi mkrTal ?-s). gamovakeli 4 da sxvaoba miviRe 9 

(rombSi Caiwereba 9, moswavleebi ki Caweren msr-s marcxena 

sqemaSi 125.2). vin gamoicnobs _ ra ricxvi Camifiqrebia? 

(daadebs TiTs ?-s). lali, ucqire am naxazs da srulad 

waikiTxe amocanis piroba! kargi, axla amovxsnaT amocana. ra 

ricxvi unda eweros aq? (roca moswavle upasuxebs, mkrTal 

?-s zemodan gadaawers 13-ians). rogor gaige? {albaT ase: 13-s 

unda gamovakloT 4, raTa miviRoT 9}. 





pasuxis mxolod Semowmebas! _ vinmes sxvagvarad xom ar gaugia 

pasuxi? {9+4=13} yoCaR, Zalian kargia!

- 311 - 

Semdeg masw.-i klass ukarnaxebs erT amgvarsave amocanas 

(raRac ricxvis Cafiqrebasa da misTvis gamoklebaze), mis 

monacemebs Seitanen msr-s meore sqemaSi (125.2), xolo 

amoxsniT _ saSinao davalebaSi amoxsnian. 

236/4/! npoblwfUfcjt! {pnwb! gvsdfm{f-! qbubsb! tbyb{bwfcjU-! 

vlwf! 31! tn.jt! gbshmfcTj/! masw.-i klass gadaaSlevinebs msr-s 

sxvadasxva naxazebs (romlebic winaswar unda hqondes 

SerCeuli!) da gaazomvinebs monakveTebs, marTkuTxedTa sigr- 

Zeebsa da sxva. yvela sigrZe santimetrebis mTel raodenobas 

unda Seicavdes, milimetirebiani naSTis gareSe! 


oRond ukve 29-is farglebSi. 

g 126. Tfebs/!3:.jt!gbshm/!nsbwbmlvUy/!hwfse/!hb{pnwb/! 

237/2/! sjdywfcjt! Tfebsfcb! 3:.jt! gbshmfcTj/! vupmpcbUb! 

Dbxfsb/!sjdywUb!sjhj;!2-!3-!4-!///!-!2:-!31-!32-!33-!///!eb!btf!Tfnefh/!! 

_ moifiqreT, romelia ufro meti _ 13 Tu 9? gamodi da 

dasvi dafaze Sesabamisi raodenobis wertilebi! (moswavle 

dasvams `dominosebur~ cameteulsa da cxraeuls} romelia 

meti? saidan ici? {13 metia, vidre 10, 9 ki naklebia, vidre 

10} davweroT: 912. 

_ moifiqreT, romelia ufro meti _ 19 Tu 21? gamodi da 

dasvi dafaze Sesabamisi raodenobis wertilebi! (moswavle 

dasvams). romelia meti? saidan ici? {21 metia, vidre 20, 19 

ki naklebia, vidre 20} davweroT: 19

- 312 - 

_ naSTia 6, aq ki naSTia 3, naklebia}. davweroT: 26>23. 

swrafad Sedardeba erTmaneTs sxva ricxvTa wyvilebi 

(oRond, aRaraa saWiro dafaze `dominoebi~ da dasabuTeba) da 

Sedarebis Sedegebi Caiwereba msr-Si 126.1. moswavleebi rigrigobiT 

waikiTxaven ricxvebs siidan, swrafad Seadareben da 

maT Soris Caweren saTanado niSans. 126.1 Sesruldeba wiTel 

xazamde, danarCeni _ saSinao davalebad miecemaT (126.7). 

237/3/!sjhpcjUj!sjdywfcj!nfUfsUnfufebo!nfpdfnef/!nbUj!eb. 

lbwTjsfcb! obtxbwm! tjwsdjU! eb! espjU! njnbsUfcfcUbo/! ! SekiTxvebiT 

ganvixilavT mTw-s mTliani ydis suraTs. SekiT-xvebis 

nimuSebi: _ vinaa rigSi meTerTmete? ra raodenobiani droSa 

ukavia mas? mecxramete? ra raodenobiani droSa ukavia mas? 

meoce? ra raodenobiani droSa ukavia mas? ocdameerTe? ra 

raodenobiani droSa ukavia mas? meTxuTmete? (da ase Semdeg). 

Semdeg daismis imdagvarive SekiTxvebi, rogorebic iyo 87.2- 

Si, oRond rigobiTi ricxvebis monawileobiT. magaliTad: _ 

mecamete ufro adre Caivlis did pepelasTan Tu meTxuTmete? 

da sxva... 

237/4/! psj! nsbwbmlvUyfejt! hwfsefcjt! hb{pnwb/! Tfefhfcjt! 

Dbxfsb!dysjmTj/!Sesruldeba msr-s 126.3. 

aq erTi siaxlea _ cifrebi gamoyenebulia Wdeebad. amitom 

saWiroa mcire Semamzadebeli saubari. _ ricxvebi risTvis 

gvWirdeba? {dasaTvlelad, Sesakrebad...} ricxvi aRniSnavs 

raodenobas: erTi cali, ori cali, sami cali... magram 

amocanebs rom win uweria ricxvebi, ai, magaliTad, es 

(miuTiTebs msr-Si) raodebobas aRniSnavs? ramdeni calia 

amocana, imas aRniSnavs? {ara} aRniSnavs amocanis nomers, ise ki 

Tqven es nomeri ar gWirdebaT. kidev sad gvxvdeba nomrebi? 

{saxlis nomeri, telefonis nomeri...} telefonis nomeri Tu 

aRniSnavs raodenobas? {ara} nomeri raodenobas ar aRniSnavs. 

_ axla daakvirdiT am mravalkuTxedebs (msr 126.3). ratom 

aweria cifrebi mravalkuTxedTa gverdebTan? gverdebis 

raodenobas gviCvenebs? {ara} gverdebis sigrZeebs xom ar 

gviCvenebs? SeamowmeT! {ara} es gverdebi gadanomrilia, raTa

- 313 - 

maTi sigrZeebi CavweroT cxrilSi. 

moswavleebs Tavidan SeiZleba cota mixmareba dasWirdeT, 

magram male gaerkvevian da Tavad daasruleben amocanas. 

237/5/!Uwmb!eb!vlvUwmb!3:.jt!gbshmfcTj/!!!_ axla daxedeT 

isev im naxazs (125.8). daiTvaleT 0-dan 29-mde! axla ukuRma 

daiTvaleT, 29-dan 0-mde! 

sakmarisia TiTo-TiTojer. Semdeg iwyeba TamaSi. wesi aseTia: 

pirveli moswavle iwyebs Tvlas. masw.-is taSis Semokraze is 

wyvets Tvlas, agrZelebs momdevno moswavle _ masw.-is Semdeg 

taSis Semokvramde, da ase Semdeg. uceb masw.-i daiZaxebs: _ axla 

ukuRma! da momdevno moswavlem ukuTvla unda ganagrZos 

zustad im ricxvidan, romelzedac wina moswavles Seawyvetina 

masw.-ma. da ase Semdeg... 

g 127. hbnfpsfcb/!ufyjmjt-!nsbwbmlvUy/!npoblwfUfcjt!hb{pnwb/! 

237/8/! _ ratom aweria cifrebi mravalkuTxedTa gverdebTan? 

gverdebis raodenobas gviCvenebs? {ara} gverdebis sigrZeebs 

gviCvenebs? {ara} aba ras gviCvenebs? {es gverdebi gadanomrilia, 

raTa maTi sigrZeebi CavweroT cxrilSi}. 

238/2/! 31.jt! gbshmfcTj! Tflsfcb.hbnplmfcjt! obtxbwmj! nbhb. 

mjUfcjt! )`bUfvm{f! hbebtwmjU~*! djgsvmj! Dbxfsb! eb! hbnf. 

psfcb/!!masw.-i asaxelebs Sekreba-gamoklebis naswavl magaliTs, 

xolo moswavleebi rig-rigobiT zepirad pasuxoben da Semdeg 

yvelani erTad weren msr-Si (cariel badeSi 127.1). savarjiSoebi 

grZeldeba manam, sanam bade mTlad ar Seivseba. 

nimuSebi: 11 _ 4; 5 + 9; 20 _ 10; 10 + 7; 14 _ 8; 9 + 

6; 19 + 1 da sxva. 

238/3/!Ubwj!24/3.jt!hbnfpsfcb!eb!Tfkbnfcb/ 

zepirad amoixsneba ori amocana, naswavli saxisa. Tuki 

moswavleebs gauWirdebaT, dafaze daixazos saTanado sqema. 

1) dodom magidaze 8 kubi daawyo. maswavlebelma aiRo 

ramdeniRac kubi, ris Semdegac magidaze 2 kubi darCa. ramdeni 

kubi auRia maswavlebels? 

2) bavSvs dedam da mamam aCuqes rveulebi, sul 8 cali.

- 314 - 

mamam 6 rveuli aCuqa. ramdeni uCuqebia dedas? 

238/4/! ufyjmfcjtb! eb! nsbwbmlvUyfefcjt! Tfnbehfofmj! npobl. 

wfUfcjt! hb{pnwb/! Tfefhfcjt! Dbxfsb! dysjmTj/! nsbwbmlvUyfejt! 

zwfmb!hwfsejt!tjhs[fUb!kbnjt!hbnpUwmb-!spdb!zwfmb!ft!tjhs[f! 

tboujnfu.sfcjt!nUfm!sjdywt!Tfjdbwt/! 

Sesruldeba 127.3. albaT, axsna ar gaxdeba saWiro. 

238/5/! `txsbgj! qbtvyfcj! `! Dbwjgjrsf! sjdywj~/ _ Cavifiqre 

ricxvi, gamovakeli 6, miviRe 6. ra ricxvi Camifiqrebia? anda: 

mivumate 3, miviRe 12. (da ase Semdeg _ naswavl magaliTebze, 

yvela _ ucnobi saklebisa an Sesakrebis moZebnaze). 

g 128. Tfls/.hbnplm/!31.jt!gbshmfcTj/!U/!23/4.jt!hbnfpsfcb/!! 

238/8/! _ ratom aweria cifrebi texilebis monakveTebTan? 

monakveTebis raodenobas gviCvenebs? {ara} maT sigrZeebs gviCvenebs? 

{ara} aba ras gviCvenebs? {es monakveTebi gadanomrilia, 

raTa maTi sigrZeebi CavweroT cxrilSi}. 

239/2/!Tflsfcb.hbnplmfcb!31.jt!gbshmfcTj!`bUfvm{f!hbebt.! 

wmjU~-! tbUbobep! DbobxfsfcjU/! yvelaferi iseve, rogorc 127.1- 

Si, oRond SigadaSig gamoereva ramdenime sxvagvari magaliTic: 

10+8, 13_3, 20+5, 26_6, 11_2, 21_2. 

------------------------------------------------------------------------------------- 

II-14. 21.ebo!31.nef!sjdywfcjt!ebTmb!TftblsfcUb!kbnfcbe!`! 

zwfmb! Tftb[mfcmpcjt! bnpxvswjU/ saerTo Ziebis Semdeg dafaze 

Camoiwereba yvela SesaZlebloba tolobaTa saxiT. klasi Tan 

kiTxulobs. 

sjdywfcjt!ebmbhfcb!nbufcjt!bo!lmfcjt!njyfewjU!)bsbvnfuft! 

8.9!dbmj!sjdywjtb*/ 

------------------------------------------------------------------------------------- 

239/3/!21.jt-!22.jtb!eb!23.jt!zwfmbobjsj!ebTmb!TftblsfcUb!kb. 

nfcbe/!!mTw gadaiSleba `10~-ze. _ radara sxvadasxva Sesakreb- 

Ta jamis tolia 10? ar gamogrCeT arcerTi SesaZlebloba! 

yvela tolobas masw.-i moswavleTa karnaxiT dafaze dawers, 

Semdeg ki klasi Seavsebs 128.2-is zeda sqemis cariel 

ujredebs. maTSi Caiwereba Tanmimdevrulad: 9+1, 8+2, 7+3,

- 315 - 

6+4 da 5+5. _ kargi, magram danarCenebi sadRa CavweroT? ai, 

magaliTad, 4+6 rom ar Cagviweria!? {4+6 tolia 6+4-isa, 

amitom aRar gvinda misi calke Cawera}. ras gviCvenebs es 10iani? 

{rom yvelas jamia 10} axla wavikiTxoT yvela jami! {10 

udris 9+1, 10 udris 8+2 ... da ase Semdeg}. 

Semdeg mTw gadaiSleba `11~-ze da klasi aseve Seavsebs 

128.2-is meore sqemis cariel ujredebs. am jamebsac erTianad 

waikiTxaven. 

bolos mTw gadaiSleba `12~-ze da klasi aseve Seavsebs 

128.2-is mesame sqemis cariel ujredebs. am jamebsac erTianad 

waikiTxaven. 

239/4/!Ubwj!23/4.jt!Tfkbnfcb!eb!hbnfpsfcb/!!128.3-is cariel adgilas 

masw.-is karnaxiT rig-rigobiT daxazaven sam nakvTs: 

1) samkuTxeds, romlis erTi gverdis sigrZea 2 sm, meorisa 

_ 4 sm (mesame _ rogoric gamova); 

2) xuTkuTxeds, romlis erTi gverdis sigrZea 1 sm, meorisa 

_ amdenive, mesamisa _ 2 sm, meoTxisa _ 1 sm (mexuTe _ 

rogoric gamova); 

3) gaxsnil texils, romlis erTi gverdis sigrZea 1 sm, 

meorisa _ 3 sm, mesamisa _ amdenive, meoTxisac _ amdenive, 

mexuTisa _ 2 sm, meeqvsisa _ 1 sm, meSvidisa _ amdenive da 

mervisa _ 2 sm. 

239/5/!Uwmb!eb!vlvUwmb!3:.jt!gbshmfcTj/ ix. 126.4, oRond 

yovelgvari Sesavlebis gareSe. 

g 129. 24.jt-!25.jt!ebTmb!kbnfcbe/!bnpdbofcj!np[sbpcb{f/! 

23:/2/! 21.ebo! 31.nef! sjdywfcjt! djgsvm! DbobxfsUb! ljefw! 

fsUyfm! eblbwTjsfcb! sbpefopcjt! bUfvmfcbe! ebUwmjtbt! 

obTUfcUbo/!masw.-i dafaze dawers 14. _ waikiTxe es ricxvi! ras 

giCvenebs es 1-iani (miTiTebiT)? {rom 14-Si 1 aTeulia} es 4iani? 

{rom aTeulTa garda kidev 4-ia} ramdeni aTeulia 14-Si? 

{erTi} risi tolia 14 _ 10? 

amdagvarive SekiTxvebiT waikiTxeba da gairCeva ricxvebi da

- 316 - 

saTanado jam-sxvaobani: 24 da 24_20; 10 da 10+10; 18 da 

18_10; 29 da 29_9; 20 da 20_10. 

_ ras gviCvenebs es 1-iani? {18-Si rom 1 aTeulia} es 8iani? 

{aTeulTa garda kidev rvaa naSTi} aq ras gviCvenebs es 

2-iani? {29-Si rom 2 aTeulia} es 9-iani? {aTeulTa garda kidev 

cxraa naSTi} Tu mixvdebiT, ratom maincadamainc aTeulebad 

daTvales es raodenobani? ratom ar SeiZleboda, magaliTad, 

rom cxraeulebad an eqvseulebad daeTvalaT? aq daewereT is, 

Tu ramdeni eqvseulicaa raodenobaSi, aq ki daeweraT is, Tu 

risi toli iqneboda eqvseulTa garda morCenili naSTi! ratom 

swored aTeulebad vTvliT yvela raodenobas? 

es sakiTxi mxolod Zlier moswavleTaTvisaa gankuTvnili. 

amitom misi ganvrcoba araa saWiro. pasuxia: _ saTvlelad yvelaze 

kargi _ TiTebia. isini yovelTvis Tan gvaqvs. raki adamians 

xelebze aTi TiTi aqvs, amitom iTvlian aTeulebad. eqvsi 

TiTi rom gvqonoda, maSin eqvseulebad daTvlidnen, rva TiTi 

rom gvqonoda _ rvaeulebad daTvlidnen. magram raki aTi TiTi 

gvaqvs, amitomaa, rom raodenobas aTeulebad viTvliT. 

23:/3/!24.jtb!eb!25.jt!zwfmbobjsj!ebTmb!TftblsfcUb!kbnfcbe/ 

yvelaferi gakeTdeba iseve, rogorc 128.2-Si, oRond 13-isa 

da 14-isaTvis. 

------------------------------------------------------------------------------------ 

III-13. vbewjmftj!bnpdbofcj!nbo[jmtb!bo!np[sbpcb{f/ 

am amocanebSi manZilebi ikribeba (manZilis adiciurobis Tviseba). 

maTi amoxsnisas saWiroa TvalsaCino sqemebis daxatva. amocanebs, 

rogorc yovelTvis, ara mxolod vxsniT, aramed winaswar kargad 

vamuSavebT da pirobas viazrebT, amoxsnis Semdeg ki pasuxsac viazrebT. 

am safexurze es xdeba sagangebo TvalsaCino sqemebis meSveobiT. 

iseve rogorc meore xazSi (ix. II-11) moswavleebma unda iswavlon 

sqemebis Sevseba, Sedgena, maTgan monacemTa amokiTxva. amitom 

amocanis amoxsnisas amocanis piroba jer sqemaze gadaitaneba, Semdeg 

ki moswavle am sqemidan aRadgens amocanis srul pirobas. 

vimeorebT da vajamebT nakvTebis naswavl dajgufebebs mTw-Si

- 317 - 

mocemuli sqema-suraTis mixedviT. 

mphjlvsj!trfnfcj!njnbsUfcb{f!`{phbej \!lfs[p~/!logikuri 

zogadi erTia da aerTianebs mraval kerZos (magaliTad, `gareuli 

cxoveli~ aerTianebs daTvs, jixvs, kiborCxalas, siraqlemas, Tevzs, 

xoWos, bus...). amiT zogadi hgavs jams. amitom logikur sqemebSi 

zogadis Sesatyvisi termini iwereba xolme elifsSi iseve rogorc 

ariTmetikis TemaSia jamebi. masw.-ic `jamad~ moixseniebs am elifss, 

xolo kerZoebis Sesabamis marTkuTxa CarCoebs ùjredebad~. 

------------------------------------------------------------------------------------- 

23:/4/!vbewjmftj!bnpdbofcj!nbo[jmtb!bo!np[sbpcb{f-!spnfmUb! 

bnptbytofmbe! tbXjspb! fsUj! fsUfvmjU! hb{pnjm! nbo[jmUb! 

Tflsfcb/! dafaze daixazeba saTanado sqema da amoixsneba amocanebi. 

sasurvelia sqemebi namdvili zomebiT daixazos (dm-Si). 

1) xoWo miwidan xis didi 

totis kenweromde abobRda. __ totis kenwero 

miwidan xis totis Ziramde 

5 dm + 4 dm 

manZilia 5 dm, totis Ziridan 

__ totis Ziri 

totis kenweromde _ 4 dm-iT 

meti. ra manZili gauvlia 

5 dm 

xoWos? 

__ miwa 

sqema martivia _ xazovani: 

gzis TviTeul monakveTs zed daewereba saTanado sigrZe. 

Semdeg _ Seikribeba. cxadia, yvelaferi moswavleTa karnaxiT, 

SekiTxvebiT gakeTdeba. _ giorgi, aq ra sigrZe unda davawero? 

ratom? aq? axla xviCam gviTxras: ratom weria aq 5 dm? 

{imitom, rom esaa manZili miwidan totis Ziramde). ratom 

weria aq 5 dm + 4 dm? {imitom, rom es manZili 4 dm-iT metia, 

vidre manZili miwidan totis Ziramde). kargi, axla dali, 

ucqire am naxazs da srulad Camoayalibe amocanis piroba! 

kargi, axla amovxsnaT amocana. am naxaziT pirdapir Cans, 

rogor unda amovxsnaT amocana, sul advilia. risi tolia 

manZili miwidan kenweromde, ai es manZili? {5 dm + 9 dm}.

- 318 - 

ciurim gviTxras amocanis sruli pasuxi! 

2) xvliki wyaros napiridan jer xemde micocda, Semdeg 

xidan _ kldis napralamde. wyarodan xemde manZilia 6 dm, 

xidan napralamde manZili amaze metia 5 dm-iT. ra manZili 

gauvlia xvliks? 

sqema martivia _ xazovani: 

6 dm 6 dm + 5 dm 

–––––––––––––– ––––––––––––––––––––––– 

wyaro xe klde 

meore amocanas iseve amoxsnian, oRond msr-Si (129.3). 

sadac daxazulia sqemisaTvis momzadebuli ori monakveTi. 

moswavleebma mxolod saTanado sigrZeebi unda daaweron, 

Semdeg ki amoxsnan amocana. 

23:/5/! Uwmb! eb! vlvUwmb! fsUjt! hbnpupwfcjU/ yvelaferi 

iseve, rogorc Tvlisa da ukuTvlisas, oRond unda gamotovon 

TiTo ricxvi. daiwyeben erTidan, unda daasaxelon mxolod 

kenti ricxvebi (cxadia, am sityvis gareSe!). 

23:/9. kanonzomierebaa: yvavili, xili, mravalkuTxedi ... 

g 130. hbnfpsfcb/!sjd/!ebmbhfcb!nbufcjU/!bn/!nbo[/!hbnplmfcb{f 

241/1/ TfojTwob/ am amocanaSi bavSvs SeiZleba gauWires 

kanonzomierebis sityvieri Camoyalibeba da sakuTari amoxsnis 

dasabuTeba. SeiZleba `yvavili~ da `xili~ ar iyos ase 

erTmniSvnelovnad garkveuli. yvavilTa da xilTa mwkrivi 

bavSvma SeiZleba gaaerTianos, an ganagrZos ase: xe, foToli, 

buCqi, balaxi... puri, yveli... amgvari pasuxebic marTebulad 

unda CaiTvalos. klasSi moswavleebi gaigeben, marTebulad 

amoxsnes Tu vera, masw.-i ki dasabuTebas nu moiTxovs. Tuki 

TviTon moswavleebi wamoiwyeben kamaTs an msjelobas, masw.-ma 

ubralod aRawerinos mwkrivi _ mis wevrTa CamoTvliT. 

amrigad, saWiroa mxolod aRwera, da ara axsna. 

241/2/!Ubwj!24/2.jt!Tfkbnfcb!eb!hbnfpsfcb/!jer gakeTdeba igive,

- 319 - 

rac 129.1-Si, oRond sxva ricxvebisaTvis: 11 da 10+1; 26 da 

20+6; 22 da 22_2; 12 da 12_10; 25 da 25_20. 

Semdeg gavameorebinebT Sekrebisa da gamoklebis naswavl 

aTiode magaliTs. visac gauWireba, mTw daixmarios. 

241/3/! sjdywfcjt! ebmbhfcb! nbufcjt! njyfewjU/! masw.-i dafaze 

dawers ricxvebs: 5, 13, 15, 20, 25, 8, 16, 0, 19, 2. 

_ es ricxvebi unda davalagoT matebis mixedviT. ese igi, 

ise unda davalagoT, rom jer eweros yvelaze patara ricxvi, 

Semdeg ki ricxvebi sul imatebdes da imatebdes. maS, romeli 

ricxvi unda davweroT sul pirvelad? {yvelaze patara} romelia 

yvelaze patara? {0} davweroT, am rigSi ki wavSaloT, rom aRar 

SegveSalos. Semdeg? darCenilTagan romelia yvelaze patara? {2} 

davweroT, zemoT ki wavSaloT. Semdeg? (da ase Semdeg): 

0, 2, 5, 8, 13, 15, 16, 19, 20, 25. 

Semdeg masw.-i dafaze dawers sxva ricxvebs: 

10, 6, 11, 27, 1, 22, 18, 14, 20. moswavleebma Tavad unda 

daalagon isini matebis mixedviT da Caweron msr-Si (130.2-is 

cariel badeSi). oRond, rom ar SeeSaloT, yoveli morigi 

ricxvi winaswar xmamaRla Tqvan. 

241/4/!vbewjmftj!bnpdbofcj!nbo[jmtb!bo!np[sbpcb{f-!spnfmUb! 

bnptbytofmbe! tbXjspb! fsUj! fsUfvmjU! hb{pnjm! nbo[jmUb! 

hbnplmfcb/! iseve rogorc 133.3-Si, imdagvarive sqemiT 

damuSavdeba da amoixsneba ori amocana, pirveli _ dafaze, 

meore _ msr-Si (130.3). 

1) magidis kididan kedlamde 18 dm manZilia. maT Soris 

dgas moswavle, magidis kididan 13 dm manZilze. ra manZilia 

moswavlesa da kedels Soris? 

18 dm 

13 dm ? dm 

––––––––––––––––––– ––––––––––––––– 

magida moswavle kedeli 

TviTeul monakveTs zed daewereba saTanado sigrZe. Semdeg 

_ gamoakldeba. cxadia, yvelaferi moswavleTa karnaxiT,

- 320 - 

SekiTxvebiT gakeTdeba (_ aq ra sigrZe unda davawero? da ase 

Semdeg _ iseve, rogorc 129.3-Si). 

2) moswavlem furcelze 19 sm sigrZis monakveTi daxaza. 

Semdeg monakveTize misi marcxena bolodan 11 sm manZilze dasva 

wiTeli wertili. ra manZilzea es wertili monakveTis 

marjvena bolodan? 

241/5/! `sjdywUb! ebnxlsjwfcb~/ igivea, rac 49.4, oRond 

mos-wavleebs urigdeba moZravi cifrebi 0-dan 9-mde, sam-sami 

cali, da maT unda moawyon ricxvTa mwkrivi 0-dan 29-mde. 

23:/9. kanonzomierebaa: ricxvi (romelic 1-iT imatebs) da 

amindis niSani, oTxidan erTerTi: Rrubeli (moRrubluloba), 

mze-Rrubeli, wvima, mze (kargi dari) _ meordeba ucvlelad. 

g 131. Tfls/!`bUfvm{f!hbebvtwm/~/!sjdyw/!ebmbh/!lmfcjU/! 

242/1/ 130.7-Si amocana gvekiTxeba, Tu ramdeni kubia 

daxazuli. ese igi, ramdeni kubi Canso am naxazze _ da ara 

imas, Tu ramdeni kubi eqneboda amgvar sxeulso. ese igi, 

moswavlem unda daTvalos mxolod is kubebi, romlebic 

uSualod moCans naxazze {21 kubia}, xolo ukan ra xdeba _ ar 

viciT da amjerad arc gvainteresebs. 

------------------------------------------------------------------------------------ 

I-14. Tflsfcb.hbnplmfcb! pdjt! gbshmfcTj! `bUfvm{f! hbeb. 

vtwmfmbe~! `! epnjoptfcsj! UwbmtbDjopfcjU! )bewjmjb*/! pdjt! 

ebTmb! TftblsfcUb! kbnfcbe! `! `pdfvmjt! UwbmtbDjopfcb{f~! 

ebnzbsfcjU/! sbnefoj! blmjb! sjdywt! bUbnef/! sbnefoj! blmjb! 

sjdywt!pdbnef/ 

------------------------------------------------------------------------------------ 

242/2/! Tflsfcb! `bUfvm{f! hbebvtwmfmbe•;! 10+4, 10+8, 11+5, 

13+6 eb!njtU/!TjhbebTjh!hbnpfsfwb!21+5, 28+1!!eb!bnebhwbsj! 

sbnefojnf!nbhbmjUj/!!!visac gauWirdeba, mTw daixmarios. 

_ ramdenia aTs mivumatoT oTxi, aTs mivumatoT eqvsi, aTs mivumatoT 

Svidi, aTs mivumatoT rva, aTs mivumatoT erTi, aTs mivumatoT 

cxra, aTs mivumatoT sami? ramdenia aTisa da erTis jami? 

_ axla SevkriboT 15 da 1. gamovides manana da dafaze

- 321 - 

dasvas 15 wertili! aTi wertilia da kidev ramdenia? {aTi da 

xuTi} daumate kidev erTi wertili! ramdenia erTi da xuTi? 

{eqvsi} sul ramdeni wertilia? {aTi da eqvsi, ese igi Teqvsmeti} 

maS, risi toli yofila 15+1? gaimeoros Salvam! 

_ ramdenia aTi da sami? {cameti} daxazos gigim! aTs da sam 

wertils, ese igi camet wertils davumatoT oTxi. ramdenia? 

(aTs da sam wertils daemateba oTxi wertili} aTs da sams 

oTxi rom davamateT, ramdeni gaxda? (pasuxi marTebuli iqneba). 

klasSi yovelTvis aris moswavle, romelic gvian xvdeba, an 

uWirs damaxsovreba. aseTi moswavleebi dafaze Tvlian 

wertilebs da ise pasuxoben. 

_ ocs da xuTs, ese igi, ocdaxuTs mivumatoT ori. ana, 

gamodi da ocdaxuTi wertili dasvi! ramdeni wertilia 

dafaze? {oci da xuTi} ramdenia ocdaxuTs mivumatoT ori? aba 

daumate kidev ori wertili! (da ase Semdeg). 

242/3/! sjdywfcjt! ebmbhfcb! lmfcjt! njyfewjU/! yvelaferi 

gakeTdeba iseve, rogorc 130.2-Si, oRond dafaze daiwereba 

sxva ricxvebi, romlebic dalagdeba klebis mixedviT. Semdeg 

daiwereba sxva ricxvebi, romlebsac Tavad moswavleebi 

daalageben klebis mixedviT da Caweren msr-Si (131.2). klebis 

mixedviT dalagebaze masw.-i ityvis: _ pirvelad unda davweroT 

yvelaze didi ricxvi, Semdeg momdevnoebi ise, rom ricxvebi 

sul iklebdes. jer yvelaze didi, mere _ darCe-nilebSi 

yvelaze didi da ase Semdeg. 

242/4/! oblwUfcjt! obtxbwm! ebkhvgfcbUb! tbfsUp! hbnfpsfcb! eb! 

Tfkbnfcb! `! UwbmtbDjopfcjt! xjhoTj! npdfnvmj! trfnb.tvsbUjt! 

njyfewjU/!mTw gadaiSleba geometriul TvalsaCinoebaze da SekiTxvebiT 

dawvrilebiT gairCeva yvela jgufi da qvejgufi. 

SekiTxvaTa nimuSebi: _ radara did jgufebadaa dayofili nakv- 

Tebi? {wertilebi, xazebi, wreebi, rgolebi, mravalkuTxedebi 

da sxvebi} amaTgan romelia kidev dayofili jgufebad? {xazebi 

da mravalkuTxedebi, sxvebi _ ara} radara jgufebadaa 

dayofili xazebi? mravalkuTxedebi? romeli mravalkuxedebis 

jgufia, romelSic SigniT sxva jgufebia? {oTxkuT-xedebi}

- 322 - 

romeli jgufebia SigniT? {markuTxedebi, marTkuTxedebSi ki _ 

kvadratebi} ratomaa, rom oTxkuTxedebis jgufSi SigniT sxva 

jgufebia? {imitom, rom yvela kvadrati _ Tan marTkuTxedicaa; 

xolo yvela marTkuTxedi _ Tan oTxkuTxedicaa} aq aris Tu 

ara daxazuli yvela SesaZlo samkuTxedi? {ara} yvela SesaZlo 

mrudi? {ara} yvela SesaZlo wre? {ara} rogori mravalkuTxedebi 

Sedis jgufSi `sxva mravalkuxedebi? {TerTmetkuTxedebi, 

TormetkuTxedebi, cametkuTxedebi da ase Semdeg} 

rogori nakvTebi Sedis jgufSi `sxvebi~? {uswormasworoebi da 

sxva yvelanairi} Tu moxvdeba am jgufSi ockuTxedi? {vera} 

ratom? {imitom, rom ockuTxedi _ mravalkuTxedia, igi mravalkuTxedebSi 

moxvdeba} romel jgufSi moxvdeba ockuTxedi? 

{sxva mravalkuTxedebis jgufSi} ockuTxedis carieli sazRvari 

romel jgufSi moxvdeba? {Sekruli texilebis jgufSi} 

wris sazRvari? {Sekruli mrudebis jgufSi} kubi Tu moxvdeba 

sadme? {ara, kubi araa nakvTi, aq ki mxolod nakvTebia}. 

_ axla Tqven TviTon dasviT axali SekiTxvebi am 

dajgufebis Sesaxeb. aba, vin rogor SekiTxvas moigonebs? 

242/5/! Uwmb! eb! vlvUwmb! fsUjt! hbnpupwfcjU/ yvelaferi iseve, 

rogorc 129.1-Si, oRond daiwyeben nulidan da unda daasaxelon 

mxolod luwi ricxvebi (cxadia, am sityvis gareSe!). 

242/9/ kanonzomierebaa: cifri (romelic 1-iT imatebs), nakvTi, 

aso, nivTi, da ase Semdeg. garda amisa, ix. SeniSvna 130.0-Si. 

g 132. `sjdywj~!eb!`djgsj~/!tjhs[/!Dbxfsb!nbufcjU!bo!lmfcjU/ 

243/2/!ufsnjofcj!`sjdywj~!eb!`djgsj~!)nbsujwbe*/!hbnfpsfcb;! 

sbnefoj!blmjb!sjdywt!bUbnef!`!zwfmb!Tftb[mfcmpcjt!bnpxv. 

swjU/!_ ai me dafaze davwer 9. ra davwere? {cxra} diax, amas, 

dawerils ewodeba cifri. kidev romeli cifrebi viswavleT? 

{0, 1, ..., 8 _ daiwereba dafaze} eseni yvela _ cifrebia. sul 

ramdeni cifria? {aTi} am aTi cali cifriT Caiwereba ricxvebi. 

sul ramdeni ricxvi gvaqvs naswavli _ aTze meti Tu naklebi? 

{meti} maT Casawerad sxva cifrebi Tu gvWirdeba? {ara, mxolod is 

aTi cali cifria sakmarisi} ricxvi _ raodenobaa, cifri ki _

- 323 - 

Canaweria. cifrebi imisaTvisaa, raTa ricxvebi CavweroT. 

_ axla naxeT es naxazi (msr-Si 132.1). aq unda CavweroT 

10-is daSla SesakrebTa jamebad ise, rom ar gamogvrCes 

arcerTi Sesakrebi, arc nuli! romeli jamebis tolia 10? 

zeda striqonSi Caiwereba: 0 1 2 3 4 5 6 7 8 9 

qveda striqonSi Caiwereba: 10 9 8 7 6 5 4 3 2 1 

Semdeg sxvadasxva moswavleebs rig-rigobiT wavakiTxebT 

Sesabamis tolobebs: 10 = 0 + 10 da ase Semdeg. 

bolos, gavameorebinebT àTeulze gadausvlelad~ Sekrebas 

(131.1-Si ganuxilavi SemTxvevebi). 

243/3/!26.jtb!eb!27.jt!zwfmbobjsj!ebTmb!TftblsfcUb!kbnfcbe/ 

yvelaferi gakeTdeba iseve, rogorc 128.2-Si, oRond 15-isa da 

16-isaTvis. 

243/4/! tjhs[fUb! Dbxfsb! nbufcjtb! bo! lmfcjt! njyfewjU/! dafaze 

daxazulia sxvadasxva sigrZeTa mqone 6 monakveTi, areulad, zogi 

ceradaa da sxva. gamova moswavle, gazomavs yvelas. Tan masw.-i 

sigrZeebs dafaze dawers (gamova areulad dawerili). Semdeg es 

sigrZeebi dalagdeba matebis mixedviT iseve, rogorc 130.2-Si. 

meorejer dafaze monakveTebi aRar daixazeba, daiwereba mxolod 

sigrZeni: 3 dm, 2 dm 5 sm, 9 sm, 3 dm 2 sm, 5 sm, 1 

dm 8 sm. es sigrZeebi dalagdeba klebis mixedviT iseve, 

rogorc 131.2-Si. Canawerebs mxolod masw.-i akeTebs dafaze. 

243/5/!txsbgj!qbtvyfcj!`!Tflsfcb!`bUfvm{f!hbebvtwmfmbe~ 

(ix. 131.1 da 132.1). 

244/9/ `logikuri kvadratebia~ (ix. ganmarteba 105.8-s win). 

marcxniv kononzomierebaa: samuSao iaraRebi, bostneuli, 

wiTeli nakvTebi. cariel ujredebSi unda exatos: raime bostneuli 

(an xili) da raime wiTeli nakvTi (sulerTia). 

marjvniv kanonzomierebaa: adamianebi, cifrebi, samgzavro 

rameebi. cariel ujredebSi unda exatos: raime cifri da 

samgzavro rame (sulerTia).

- 324 - 

g 133. hbnplmfcb!`bUfvm{f!hbebvtwmfmbe~.!!hbnfpsfcb/! 

244/2/!hbnplmfcb!`bUfvm{f!hbebvtwmfmbe~;!29`5-!24`3-!26`6-!! 

2:`28-! 27`23! eb! njtU/! TjhbebTjh! hbnpfsfwb! 36`5-! ! 39`3! eb! 

bnebhwbsj! sbnefojnf! nbhbmjUjd/! _ aTs da rvas gamovakloT 

xuTi, ese igi, Tvramets gamovakloT xuTi (Sesruldeba 

TvalsaCinod, iseve rogorc 131.1-Si). aTs da cxras, ese igi 

cxramets gamovakloT oTxi. aTs da Svids, ese igi Cvidmets 

gamovakloT sami. aTs da eqvs, ese igi Teqvsmets gamovakloT 

oTxi. ocs da xuTs, ese igi ocdaxuTs gamovakloT ori. ocs 

da rvas, ese igi ocdarvas gamovakloT sami. 

yovel pasuxze masw.-ma unda moiTxovos axsna. sasurvelia, 

moswavle aseT pasuxamde miviyvanoT: Cvidmeti aris aTi da 

Svidi. Svids gamovakloT sami udris oTxs. aTi da oTxi ki 

tolia ToTxmetisa. maS, Cvidmets gamovakloT sami _ darCeba 

ToTxmeti. 

aseve Sesruldeba 19–7 da misTanani. 

sxva mxriv iseve, rogorc 131.1-Si. 

244/3/!28.jtb!eb!29.jt!zwfmbobjsj!ebTmb!TftblsfcUb!kbnfcbe/ 


da 18-isaTvis. 

244/4/!Ubwj!24/4.jt!Tfkbnfcb!eb!hbnfpsfcb/! swrafad amoixsneba 

imdagvarive amocanebi, rogorebic iyo 129.3-Si da 130.3-Si. 

dafaze daixatos sqemebi. 

1) rafaze dgas yvavilebis qoTani, larnaki da saTamaSo (am 

TanmimdevrobiT). qoTansa da larnaks Soris manZilia 4 dm, 

larnaksa da saTamaSos Soris _ 8 dm. ra manZilia qoTansa da 

saTamaSos Soris? 

2) ku sorodan gamobobRda, gaiara 19 dm manZili, SeCerda 

da balaxis Wama daiwyo. Semdeg sorodan gamobobRda patara ku, 

wavida didi kusaken da gaiara jer 3 dm manZili, Semdeg kidev 

amdenive. ra manZili darCa did da patara kuebs Soris?

- 325 - 

244/5/! `txsbgj! qbtvyfcj! `! Tfnefhjt! Tfnefhj-! xjojt! xjob~/ 

masw.-i asaxelebs ricxvs (magaliTad, 15), moswavlem unda 

daasaxelos am ricxvis Semdegis Semdegi ricxvi {17}. meore 

wreze _ am ricxvis winis wina ricxvi {13}, mesame wreze _ 

Semdegis Semdegis Semdegi {18}, meoTxeze _ winis winis wina {12}. 

masw.-ma ricxvebi yuradRebiT unda daasaxelos, raTa moswavles ar 

mouwios 0-dan 29-mde farglebis gareT gasvla. 

g 134. sbne/!blmjb!sjdywt!pdbnef/!Ubonjne/!tjwsd/!eb!espTj/ 

245/2/! sbnefoj! blmjb! sjdywt! pdbnef/! zwfmb! Tftb[mfcmpcjt! 

bnpxvswjU/!sqema Seivseba da waikiTxeba iseve, rogorc 132.1-Si, 

oRond ocisaTvis. Sevsebisas gamoiyeneben mTw-Si 20-is 

marTkuTxa TvalsaCinoebas (ix. gv. 17). SekiTxvebi: _ 

TerTmeti rogoraa dawyobili? {aTi da erTi} Tormeti? {aTi 

da ori} cameti? {aTi da sami} ToTxmeti rogoraa dawyobili? 

{aTi da oTxi} TxuTmeti? {aTi da xuTi} Teqvsmeti? {aTi da 

eqvsi} Cvidmeti? {aTi da Svidi} Tvrameti? {aTi da rva} 

cxrameti? {aTi da cxra} oci? {aTi da aTi} jer cxramets 

daakvirdiT, ramdeni aklia ocamde? amixseni, rogor mixvdi! 

daakvirdi Tvramets. ramdeni aklia ocamde? ese igi, Tvramets 

ramdeni unda mivumatoT, raTa oci miviRoT? (da ase Semdeg) 

TerTmets ramdeni unda mivumatoT, oci rom miviRoT? 

am gakveTilze Seivsos mxolod erTi sqema, meore _ 135.1-ze. 

245/3/!2:.jtb!eb!31.jt!zwfmbobjsj!ebTmb!TftblsfcUb!kbnfcbe/ 


da 20-isaTvis.! 

245/4/! tbfsUp! Tfkbnfcb! `! mTw.jt! zejt! ebxwsjmfcjUj! 

hboyjmwb;! lbopo{pnjfsfcjt! hbb{sfcb/! np[sbpcjtbt! xbsnprnojm! 

tjwsdjU! eb! espjU! njnbsUfcbUb! fsUespvmj! eblbwTjsfcb! 

obtxbwm! sjhpcjU! sjdywfcUbo/! Ubonjnefwspcb! tjwsdftb! eb! 

espTj/! SekiTxvebiT gairCeva mTw-s ydis suraTi mTlianad. _ 

sul ramdeni yvavilia? ramdeni pepelaa? ramdeni bavSvi da

- 326 - 

cxoveli midis bilikze? {ocdaori} ocdaorives ukavia Tu ara 

sxvadasxvanairi droSebi? {ki} droSebze ra aris gamosaxuli? 

{nakvTebi da raodenobani} axla daakvirdiT _ ra kanonzomierebas 

amCnevT _ vis romeli raodenoba da romeli nakvTi 

aqvs? vinaa sul pirveli? {biWuna} mas Tu aqvs raodenoba? 

{ara, mas aqvs droSa wignis saxelwodebiT} Semdegi vinaa? 

{gogona} mas Tu aqvs nakvTi? {ki, aqvs rgoli} raodenoba 

romeli aqvs? {araferi, arcerTi} araferi, arcerTi _ es ra 

ricxvs Seesabameba? {nuls} ese igi, gogonas hqonia nuli! xom 

hgavs nuls mTlianad misi droSa! Semdegi vin modis bilikze? 

{cuga} mas romeli nakvTi da romeli raodenoba aqvs? {wre da 

erTeuli} Semdeg? {fisoa, mas aqvs marTkuTxedi da oreuli} 

Semdeg? {mamali _ samkuTxedi da sameuli} Semdeg? {dedali, 

kvadrati da oTxeuli} (da ase Semdeg meaTemde _ ciyvi, 

aTkuTxedi da aTeuli). amis Semdeg iwyeba aTze meti 

raodenobani! magram jer aqve vnaxoT _ ra kanonzomiereba iyo? 

{raodenobani TanmimdevrobiT iyo: nuli, erTi, ori... sameuli 

iyo samkuTxa droSaze, oTxeuli _ kvadratulze, xuTeuli _ 

xuTkuTxaze...} diax, biWuna da gogona sul calke arian, 

danarCenebs, cxovelebs ki is raodenoba aqvT, meramdenec arian 

bilikze! aba, swrafad, dauTvlelad mipasuxeT _ cxovelebSi 

meramdenea bu? {meeqvse} baWia? {mecxre} mamali? {mesame} 

meSvide vinaa? {ixvis WuWuli} meoTxe? {dedali} merve? {goWi} 

ocamdec es kanonzomiereba grZeldeba. aba swrafad moZebneT: 

cxovelebSi meToTxmete vinaa? {kvici} mela meramdenea? 

{meCvidmete} kargi, axla vin moifiqrebs, biWuna da gogonac 

rom CavTvaloT, maSin meramdene iqneba mela? {mecxramete} 

ratom? {imitom, rom win orni daematnen} maSin meTxuTmete vin 

iqneba? {nukri, romelsac cameteuli ukavia xelSi}. 

Semdeg daismis imdagvari SekiTxvebi, rogorebic iyo 87.2-Si. 

magram erTi gansxvavebiT: did pepelas ki aRar SevudarebT 

Camvlelebs, aramed sxva rames: _ warmoidgineT, rom biWunam

- 327 - 

Tavisi droSa iqve CaarWo, sadac dgas. Semdeg gogona da biWuna 

win wavidnen da aRar Canan. maSin yvelaze adre vin Caivlis 

did droSasTan? {cuga} yvelaze gvian? {ku} meramdene Caivlis 

xbo? meaTe vin Caivlis? {da ase Semdeg}. meTxuTmete ufro 

winaa rigSi Tu meTvramete? bokveri ufro gvian Caivlis Tu 

cikani? meTormete vin Cavlis? 

245/5/ amdagvar SekiTxvebs miecema TamaSis saxe _ `swrafi 

pasuxebi~. 

g 135. Tflsfcb!)kbnjU!31*/!npdfn/!tjhs[jt!ufyjmjt!ebyb{wb/ 

246/1/!134.8-Si: zeda rigSi 13 unda Seicvalos 12-iT, radgan 

kanonzomierebaa `2-iT mateba~; Sua rigSi 12 unda Seicvalos 

11-iT, radgan kanonzomierebaa `2-iT kleba~; qveda rigSi 21 

unda Seicvalos 22-iT, radgan kanonzomierebaa `2-iT kleba~. 

246/2/!Tflsfcb!)kbnjU!31*;!266-!293-!284-!22:-!2:2!!eb!tywb/ 

Seivseba qveda sqema (134.1-135.1). Semdeg, kidev erTxel, 

areuli TanmimdevrobiT, kargad gavameorebinebT am friad 

mniSvnelovan jamebs. mTw gadaSlili hqoneT 20-is marTkuTxa 

TvalsaCinoebaze (gv. 17), xolo msr _ 135.1-ze. 

246/3/! Ubwj! 25/3.jt! hbnfpsfcb! eb! Tfkbnfcb/! ! yvelanair SesakrebTa 

jamebad daSlian 11-s. Sesakrebebi dafaze Camoiwereba. 

Semdeg maT daalegeben matebis mixedviT. 

yvelanair SesakrebTa jamebad daSlian 18-s. Sesakrebebi 

dafaze Camoiwereba. Semdeg maT daalegeben klebis mixedviT. 

Semdeg msr-Si ukve Sevsebuli saTanado cxrilebis 

daxmarebiT yvelanair SesakrebTa jamebad daSlian 10-s, 12-s, 

13-s, 14-s, 15-s, 16-s, 17-s, 19-s, 20-s. weris gareSe, mxolod 

rig-rigobiT waikiTxaven msr-dan. 

bolos, masw.-i dafaze dawers: 15, 3, 0, 17, 4, 20, 1. 

_ zepirad vin daalagebs am ricxvebs matebis mixedviT? kargi, 

axla klebis mixedviT! 

Ubwj!25/4; mesame xazis ZiriTadi programa dasrulebulia. Zi-

- 328 - 

riTad naswavl sakiTxTa gameoreba, ganmtkiceba da ganviTareba. 

246/4/! npdfnvmj! tjhs[jt! ufyjmjt! ebyb{wb/! msr-Si Sesruldeba 

amocana 135.3. mas cota mosazreba sWirdeba, SeiZleba, 

sust moswavleebs SeeSaloT, an ver SeZlon daxazva. maT masw.-i 

miexmareba: _ aba, rogor SeiZleba davSaloT 10 oTxi 

Sesakrebis jamad? {2+2+3+3} kargi, axla daxaze texili 

amdeni santimetris sigrZis monakveTebiT. mixvdi, romeli 

texili unda dagexaza? aba miTxari! gaamuqe! gaxsnilia es 

texili Tu Sekruli? {gaxsnili} (Sekrulis daxazva gacilebiT 

ufro Znelia!). 

_ axla me mogcemT erT mosasazrebel davalebas. vin 

daxazavs swrafad sammonakveTian texils, romlis mTliani 

sigrZe iqneba isev 10 sm? 

Zlierma moswavleebma unda moisazron: daxazon TiTqmis 

igive texili, oRond mas ori mezobeli monakveTi gauerTianon 

erT monakveTad. _ ra sigrZisaa am texilis Semadgeneli 

monakveTebi? 

246/5/! `txsbgj! qbtvyfcj! `! Dbwjgjrsf! sjdywfcj~/! am TamaSSi 

siaxle isaa, rom gamosacnobia ori ricxvi _ sakleb-maklebi an 

ori Sesakrebi. Tanac, gansxvavebiT wina TamaSebisagan, pasuxTa 

sxvadasxva SesaZleblobaa. masw.-i ambobs: _ Cavifiqre ricxvi, 

gamovakeli meore ricxvi, sxvaoba miviRe 5. romeli saklebi da 

maklebi SeiZleboda Camefiqrebina? {magaliTad, 10 da 5; anda 6 

da 1} Cavifiqre ricxvi, mivumate meore ricxvi, jami miviRe 

19. romeli Sesakrebebi SeiZleboda Camefiqrebina? {10 da 9, 

anda 15 da 4}. 

g 136. hbnplm/!`bUfvm{f!hbebvtwmfm~/!mphjl/!trfnb!njnbsU/ 

136.1. gamokleba àTeulze gadausvlelad~: 20_1, 20_2, ... , 

20_19. 

arsebiTad, es ukve iyo 134.1-Si da 135.1-Si. axla kidev 

erTxel, areuli TanmimdevrobiT, kargad gavameorebinebT am 

friad mniSvnelovan sxvaobebs. mTw gadaSlili hqondeT 20-is

- 329 - 

marTkuTxa TvalsaCinoebaze (gv. 17), xolo msr _ 134.1-ze. 

Ubwj! 26/3; meore xazis ZiriTadi programa dasrulebulia. ZiriTad 

naswavl sakiTxTa gameoreba, ganmtkiceba da ganviTareba. 

247/3/! hblwfUjmfcjt! dysjmjt! hboyjmwb/ msr-Si Sesruldeba 

amocana 136.2. moswavleebi masw.-is SekiTxvebis meSveobiT 

nawilobriv Seavseben cxrils _ sakuTari gakveTilebisas. 

Semdeg masw.-i dasvams ramdenime imdagvarsave SekiTxvas, 

rogorebic iyo 102.2 da 103.2-Si. cxrilis Sevsebas Sin daasruleben 

(136.9). 

247/4/!mphjlvsj!trfnb!njnbsUfcb{f!eb!lfs[p/{phbej/! 

iwyeba saintereso logikuri sqemebis Sevseba. es sqemebi 

aTvalsaCinoebs mimarTebas `zogadi \ kerZo~ _ ix. msr $ 8. 

_ daakvirdiT jer zeda marcxena ujredebs (msr-s 136.3). 

ra weria Sig? {bavSvi, gogona, biWuna} gogona da biWuna vin 

arian? {bavSvebi} qaTami da SaSvi? {frinvelebi} am Sua qveda 

ujredSi ra unda eweros? {mesame frinvelis saxeli, SeiZleba, 

magaliTad, mtredi an beRura} Semdeg rom sami ujredia carieli, 

iq ra unda eweros? {xilis saxelebi, magaliTad: vaSli, 

msxali, komSi, atami...} axla qveda striqonSi, marcxniv. ra 

unda eweros `jamSi~? vin arian qali da mamakaci? {adamianebi} 

ese igi, ra unda eweros cariel `jamSi`? {adamiani} Semdeg? 

{`jamSi~ _ cxoveli; ujredSi _ cxovelis saxeli, SeiZleba 

spilo, ZaRli an zvigeni} Semdeg sam ujredSi? {magaliTad: 

wre, monakveTi, samkuTxedi}.! 


oRond ukve 29-is farglebSi. 

g 137. hbnfpsfcb/!hblwfUjm/!dysjmj/!psnprnfe/!bnpd/!np[s/ 

248/2/! Ubwj! 25/2.jt! hbnfpsfcb! eb! Tfkbnfcb;! Tflsfcb.hbnplmfcb! 

pdjt! gbshmfcTj! `! zpwfmhwbsj! TfnUywfwfcj/! mimdinareobs 

zepiri Sekreba-gamokleba, yovelgvari magaliTebiT, maT Soris 

jamiT 20. magaliTebi:

- 330 - 

1) Sekreba àTeulze gadausvlelad”: 

13 + 5; 14 + 3; 15 + 5; 20 + 4; 21 + 4; 12 + 7. pirveli 

Sesruldeba dafasTan dasabaTebiT _ iseve, rogorc keTdeboda 

131.1-Si; danarCenebi _ pirdapir zepirad. 

2) Sekreba àTeulze gadasvliT”: 

8 + 5; 3 + 9; 7 + 7; 6 + 8; 9 + 4; 9 + 9. pirveli Sesruldeba 

mTw-s daxmarebiTa da msjelobiT, danarCenebi _ pirdapir 

zepirad (visac gauWirdeba, TavisTvis gamoiyenos mTw). 

3) gamokleba àTeulze gadausvlelad”: 

18 _ 4; 19 _ 6; 20 _ 8; 25 _ 4; 17 _ 6; 26 _ 6. 

pirveli Sesruldeba dafasTan dasabuTebiT _ iseve, rogorc 

keTdeboda 133.1-Si; danarCenebi _ pirdapir zepirad. 

4) gamokleba àTeulze gadausvlelad”: 

18 _ 14; 19 _ 16; 20 _ 18; 27 _ 22; 17 _ 13; 16 _ 11. 

pirveli Sesruldeba dafasTan dasabaTebiT _ iseve, rogorc 

keTdeboda 133.1-Si; danarCenebi _ pirdapir zepirad. 

5) gamokleba àTeulze gadasvliT”: 

15 _ 7; 17 _ 8; 14 _ 6; 13 _ 9; 11 _ 5; 20 _ 6. 

pirveli da meore Sesruldeba mTw-s daxmarebiTa da 

saTanado msjelobiT, danarCenebi _ pirdapir zepirad (visac 

gauWirdeba, TavisTvis gamoiyenos mTw). 

angariSi gagrZeldeba TamaSSi (137.4). 

248/3/! hblwfUjmfcjt! dysjmjt! hboyjmwb/! masw.-i dasvams 

SekiTxvebs ukve Sevsebuli cxrilis Sesaxeb (136.2.137.2), 

droiT mimarTebaTa CarTviT: _ samSabaTs romeli gakveTili 

gaqvT maTematikasa da musikas Soris? xuTSabaTs meramdene 

gaqvT qarTuli? romel dReebSi gaqvT maTematika pirvel 

gakveTilad? orSabaTs ra gaqvT maTematikis win? musikis 

Semdeg? paraskevs ris Semdegaa fizkultura? ris winaa xatva 

romelia is sagnebi, romlebic yoveldRe gaqvT? romel dReebSi 

ara gaqvT xatva? sul ramden sagans swavlobT Tqven? erT 

kviraSi sul ramdenjer gitardebaT maTematika? qarTuli?

- 331 - 

Semdeg moswavleebi erTmaneTs dausvamen amgvarsave 

SekiTxvebs cxrilis Sesaxeb. 

248/4/! TfebsfcjU! sUvmj-! psnprnfefcjboj! bnpdbob! np[sbp. 

cb{f/! iseve rogorc 129.3-Si, sqemis mixedviT damuSavdeba da 

amoixsneba amocana 137.3. moswavleebi msr-Si mocemul sqemas 

saTanado manZilebs daawren, Semdeg erTi moswavle sqemis mixedviT 

gaimeorebs amocanis pirobas, Semdeg amocana amoixsneba. 

248/5/! `txsbgj! qbtvyfcj! `! bohbsjTj~/! TamaSis saxiT 

gagrZeldeba 137.1-is magvari zepiri gamoTvlebi. 

ganmamtkicebel-ganmavaTarebeli gakveTilebi 

pirveli klasis ZiriTadi programa dasrulebulia. magram 

saWiroa misi mniSvnelovani sakiTxebis gameoreba, ganmtkiceba da 

ganviTareba aqtiurad, amocanebis meSveobiT. 

gakveTilis agebuleba icvleba: aRar iqneba sagakveTilo 

monakveTebi I, II da III xazebidan. gakveTilebi ZiriTadad amocanebs 

daeTmoba. sxva siaxlecaa: igulisxmeba, rom am droisaTvis klasma 

ukve icis kiTxva. amitom amocanaTa pirobani aRaraa vardisfer 

fonze, samagierod, msxvili SriftiTaa. yvela amocanis piroba 

moswavlem ukve Tavad unda waikiTxos da Tavadve gaiazros igi! es 

unda icodnen mSoblebmac da masw.-ma maT unda gaagebinos, rom 

daTvur samsaxurs nu gauweven bavSvebs. iSviaTad, roca amocanis 

pirobas damatebiTi axsna sWirdeba, amas Cven sagangebod aRvniSnavT 

aqve. Tuki romelime moswavles gansakuTrebiT uWirs kiTxva, 

wakiTxvaSi mas mainc ufrosi unda daexmaros Sin, meti gza araa. 

magram es iZulebiTia da Zalian arasasurveli. amoxsnaSi daxmareba 

ki mxolod Seferxebuli ganviTarebis bavSvs dasWirdeba. esec unda 

icodes mSobelma. rasakvirvelia, es imas ar niSnavs, rom yvela 

moswavle yvela amocanas uSecdomod amoxsnis. magram es araa 

arsebiTi. dae hqondeT Secdomebi, ufro metic, iwvalon da Tundac 

sul veraferi ver daweron. mTvaria is, rom moswavlem damou-

- 332 - 

kideblad unda imuSaos amocanaze! rasac SeZlebs, gaakeTebs, rasac 

vera momdevno gakveTilze garCevisas gaigebs da Seasworebs Tu 

Seavsebs. swored esaa namdvili swavlis erTaderTi gza da mas 

gaanadgurebs ufrosTa mier gaweuli daTvuri daxmareba. 

masw.-i gahyveba msr-s (romelSic amocanebi kvlav gakveTilebis 

mixedviTaa gadanomrili). magram, garda amisa, masw.-s unda axsovdes, 

rom yovel gakveTilze man damatebiT unda Caataros ori TamaSi, 

romlebic am wignSi aRar iqneba miTiTebuli. TiTo TamaSs eTmoba, 

rogorc wesi, 4-6 wuTi. 

erTi TamaSia `txsbgj! qbtvyfcj! `! {fqjsj! bohbsjTj! 31.jt! 

gbshmfcTj~: yovelgvari erTmoqmedebiani Sekreba-gamokleba 

(pirvel wreze SedarebiT advilebi, Semdeg ufro Znelebi). 

meore TamaSia: `.jU!nfuj / .jU!oblmfcj!`!kbXwvsbe~. sawyis 

ricxvs asaxelebs masw.-i, magaliTad: 0, 12, an 3, sulerTia {3}. 

pirvelma moswavlem unda daasaxelos am ricxvze 2-iT meti ricxvi 

{5}, meore moswavlem ukve 5-ze 2-iT meti ricxvi {7}, mesamem 7ze 

2-iT meti {9} da ase Semdeg, sanam masw.-i taSs ar Semokravs da 

ar daiZaxebs: axla 2-iT naklebi! 

moswavleTa mier dasaxelebuli ricxvebis udidesi zeda 

sazRvaria 29. Tumca, masw.-s SeuZlia, zogjer ufro adrec 

gawyvitos xolme mateba da TamaSi klebaze gadaiyvanos. aseve, masw.-s 

SeuZlia, nulze ufro adre gawyvitos kleba da TamaSi kvlav 

matebaze gadaiyvanos. 

meore wreze 2-iT met-nakleboba icvleba 3-iT met-naklebobiT, 

mesame wreze 4-iT met-naklebobiT da ase Semdeg. 

es ramdenime TvalsazrisiT saukeTeso TamaSia, meore klasSi igi 

Zalian xSirad iqneba. 

yovel gakveTilze pirvel TamaSad tardeba an `swrafi pasuxebi 

zepiri angariSi 20-is farglebSi~, an `-iT meti / -iT naklebi 

jaWvurad~ (dRegamoSvebiT, rig-rigobiT: an erTi, an meore). meore

- 333 - 

TamaSs zogierT gakveTilze vuTiTebT. magram umravlesobaze arcerTi 

TamaSi araa miTiTebuli. igulisxmeba, rom masw.-i pirvel 

TamaSad Caatarebs zemore aRwerilTagan erTerTs, xolo meore 

TamaSs Tavad SearCevs, sakuTari surviliT (adre mocemulTagan). 

garda amisa, erTerT ukanasknel gakveTilze unda CaerTos es 

xumroba, sasacilo ambavi. 

_ bavSvebo, Cvens skolaSi romelia yvelaze kargi niSani? 

{gaaCnia skolas: an xuTiani, an cxriani, an aTiani}. romeli 

niSnis miReba uxariaT yvelaze metad? {cxrianisa} hoda 

erTxel da-Zma Sin dabrundnen, skolidan. Sin ki Turme 

stumrebi yofilan. _ ra niSnebi miiReT skolaSi? _ hkiTxes 

bavSvebs. dam upasuxa: _ me cxriani miviRe! _ yoCaR! raSi 

miiRe? _ maTematikaSi. _ Zalian kargia, Zalian kargi gogona 

yofilxar! ai Sen saCuqari! _ da stumarma deidam gogonas 

saCuqari gadasca. _ mec ki miviRe cxriani! _ wamoiZaxa Zmam. _ 

Senc yoCaR! raSi miiRe cxriani? _ miviRe! qarTulSi miviRe 

oriani, maTematikaSic _ oriani, fizkulturaSi _ xuTiani, 

hoda erTad sul xom cxriani gamodis! _ ratom iciniT, 

bavSvebo? gana cudad iangariSa? {kargad axsnan xumrobis 

azri!} ki magram, Tuki ase kargad icoda Sekreba, maTematikaSi 

oriani ratom miiRo? (gamoiTqmeba sxvadasxva azri). 

Tuki skolaSi umaRlesi niSania xuTiani an aTiani, es ambavi 

Sesabamisad gadakeTdeba. 

gakveTili, rogorc yovelTvis, iwyeba saSinao davalebis safuZvliani 

erToblivi ganxilviT. Semdeg klasi damoukideblad amoxsnis 

msr-s Sesabamisi nomris gakveTilis pirvel, meore, mesame... amocanas 

ramdenic moeswreba. danarCenebi saSinao davalebad miecemaT. 

magaliTad, 142-e gakveTilze jer gairCeva wina gakveTilze 

micemuli saSinao davaleba: 141.5, 141.6, 141.7, 141.8 da 141.9. Semdeg 

SeiZleba Catardes pirveli TamaSi. Semdeg amoixsneba 142.1, 142.2,

- 334 - 

142.3. Semdeg Catardeba meore TamaSi. Semdeg, Tuki moeswreba, 

amoixsneba 142.6. danarCenebi (142.5-dan 142.9-mde) klass saSinao 

davalebad miecema. yvela SemTxvevaSi is amocanebi, romelTa amoxsna 

klasSi ver moeswreba, saSinao davalebad eZlevaT. 

aq Cven gavarCevT mxolod im amocanebs, romlebsac cota 

sxvagvari, Taviseburi msjeloba an axsna SeiZleba dasWirdes. 

msr-Si amocanebi isea SerCeuli, rom erToblivad isini 

mTlian programas moicavs. amitom am amocanebis amoxsna TavisTavad 

iqneba naswavlis aqtiuri gameoreba da ganviTareba. amasTan, 

amocanebi gamoavlens moswavleTa codnis xarvezebsac. cxadia, masw.ma 

es xarvezebi uyuradRebod ar unda Catovos! 

cxadia, mTlianad msr-s gavla pirvel klasSi ver moeswreba. 

masw.-ma klass sazafxulo davalebac aucileblad unda misces. es 

iqneba is amocanebi, romelTa gavlac ver moeswro saswavlo wlis 

ganmavlobaSi (SeiZleba iyos msr-s wina nawilebidanac romelic 

mizezTa gamo ar amoxsnila). amgvar amocanebs masw.-i moZebnis da 

klass Cveulebriv CaaniSninebs msr-Sive. mTlianobaSi sazafxulo 

davalebad klass unda mieces daaxloebiT 50-60 amocana. Tan 

daevalebaT, rom dReSi TiTo amocana amoxsnan, zogjer SeiZleba 

mecadineoba sul ver SeZlon, magram roca SeZleben, dReSi TiTo 

an ori amocana amoxsnan. 

sazafxulo davalebas udidesi mniSvneloba aqvs ufro meti, 

vidre Cveulebriv saSinao davalebebs. es davaleba, jer erTi, 

moswavles karg gonebriv mdgomareobas unarCunebs da xels uSlis 

naswavlis daviwyebas. aranaklebi mniSvneloba aqvs zogad aRmzrdelobiT 

zemoqmedebas. bavSvi eCveva yoveldRiur rudunebas mcire, 

Zalian advil, magram mainc gegmazomier da mizandasaxul yoveldRiur 

rudunebas. es ufro mniSvnelovania, vidre sazogadod 

maTematikis swavla. 

meore klasis dasawyisSi masw.-i moagrovebs rveulebs da

- 335 - 

gaasworebs moswavleTa sazafxulo namuSevrebs. yvela keTilsindisieri 

moswavle unda waxalisdes SeqebiTa da umaRlesi niSniT. 

meore klasis pirveli ori-sami dRe daeTmoba sayuradRebo amocanebis 

klasSi garCevas da imdagvarsave TamaSebs, rogorebic aqve 

aRvwereT. es iqneba naswavlis mokle gameoreba. amocanebs masw.-i 

aarCevs sakuTari SexedulebiT: romlebic SedarebiT gauWirdaT moswavleebs, 

romlebic masw.-s saintereso eCveneba da sxva. SedarebiT 

swrafad gairCeva imdeni, ramdenic moeswreba or-sam gakveTilSi. 

g 138. 249/2/! amocana advilia, oRond yuradRebiT dakvirveba 

sWirdeba. sanam saWiro cifrebs Caweren ujredebSi, zepirad 

unda imsjelon da axsnan pasuxi. 

erTi amgvari amocana saSinao davalebadac eZlevaT (138.6). 

249/3/ cariel adgilebSi Caiwereba saTanado ariTmetikuli 

niSnebi, raTa miviRoT marTebuli tolobani. es male gakeTdeba. 

Semdeg masw.-i yuradRebas gaamaxvilebs imaze, rom es magaliTebi 

raRac kanonzomierebiT weria. _ vin aRmoaCens kanonzomierebas? 

{zeda striqonSi Sekrebis magaliTia, mis qvemoT ki _ misi 

Sesabamisi gamoklebis magaliTi} vTqvaT, viangariSeT zeda jami. 

qveda sxvaobas Tavidan Tu sWirdeba angariSi? {ara, jami 

pirdapir gvkarnaxobs}. kargi, axla me dafaze davwer jams: 5 + 

8 = 13. axla swrafad mipasuxos daTom (saSualo moswavle), 

risi tolia 13 _ 8? avTo, risi tolia 13 _ 5? axla vin 

moigonebs amgvar axal magaliTebs Sekreba-gamoklebaze? 

249/4/ ar sWirdeba ganmartebani, moswavleebma TviTonve 

waikiTxon amocanis piroba da Seasrulon misi piroba. mTavaria 

is, rom yvelaferi zedmiwevniT zustad iqnes Sesrulebuli! 

249/5/!`txsbgj!qbtvyfcj!`!dysjmj~/!TamaSis saxiT gagrZeldeba 

137.2-is magvari zepiri SekiTxvebi cxrilisa da droiT 

mimarTebaTa Sesaxeb.

- 336 - 

g 139. 24:/1/ saguldagulo ganxilva sWirdeba 138.6-sa da 138.7-s. 

24:/2/ masw.-i jer zepir pasuxebs moiTxovs rig-rigobiT 

sxvadasxva moswavlisagan, saTanado msjelobiT. msr-s 

ujredebs (139.1) Semdeg Seavseben. Semdeg masw.-i ramdenime 

amgvar axal SekiTxvas dasvams da moswavleebi zepirad 

upasuxeben maT: _ romelia is ricxvi, 11 romelzec 4-iTaa 

naklebi? romelia is ricxvi, 9 romelzec 6-iTaa naklebi? 

romelia is ricxvi, 18 romelzec 3-iTaa meti? romelia is 

ricxvi, 20 romelzec 4-iTaa meti? (da kidev ramdenime). 

24:/3/ suraTi ukve gaferadebulia. _ ramdeni sxvadasxva 

gziT SeiZleba mivides moswavle skolaSi? yvela gza unda 

vnaxoT, grZeli da mixveul-moxveuli gzebic, yvela! 

gzebis gamartivebuli sqema daixazos dafaze. ar unda 

gamogvrCes arcerTi SesaZlebloba: 

1) zemo gzajvaredini _ zemo xidi _ skola; 

2) zemo gzajvaredini _ Sua gzajvaredini _ Sua xidi _ skola; 

3) zemo gzajvaredini _ qvemo gzajvaredini _ qvemo xidi _ skola; 

4) qvemo gzajvaredini _ zemo gzajvaredini _ zemo xidi _ skola; 

5) qvemo gzajvaredini _ Sua gzajvaredini _ Sua xidi _ skola; 

6) qvemo gzajvaredini _ qvemo xidi _ skola. 

_ kidev Tu aris sxva SesaZlebloba? {ara} ratom? imitom, 

rom moswavlem an zemo gzajvaredini unda gaiaros, an qvemo. 

mere ki zemo xidi unda gaiaros, an Sua, an qvemo. sxva gza 

araa. kargi. axla daakvirdiT: romelia yvelaze mokle gza? 

{pirveli} yvelaze grZeli? {meoTxe}. 

_ ramdeni fiWvia zeda gzis piras? ramdeni saxlia? ramdeni 

mTaa? ramdeni xidia? kidev ra moCans? ramdeni? 

24:/4/ daaxloebiT amgvari amocana ukve Segvxvda (136.3). _ 

rva aris ricxvi. xazis qvemoT ki ra SeiZleba eweros? 

{SeiZleba eweros monakveTi, SeiZleba _ texili, an _ mrudi}.

- 337 - 

SemdegSi ra weria? {wre} zemodan ra unda eweros? wre ra 

aris? {nakvTi} Semdeg? {raime aveji, magaliTad, SeiZleba merxi, 

an skami, an magida...} Semdeg? {xe} Semdeg? {bostneuli} Semdeg? 

{nakvTi}. vin moisazrebs, aq kidev ra SeiZleboda? kidev ra 

aris marTkuTxedi? {oTxkuTxedi} ai bolo ki Znelia. iseTi 

rame unda CavweroT, rac aris marTkuTxedi. sxva ramea, magram 

Tan marTkuTxedicaa, yovelTvis marTkuTxedia! {kvadrati}. 

24:/5/!`txsbgj!qbtvyfcj!`!sjdywfcjt!ubTfcj/~ ix. 122.4. 

g 140. 251/1/ iseve, rogorc 130.7-Si (ix. 131.0), 139.7-Si unda 

davTvaloT mxolod aq daxazuli kubebi _ rogorc moiTxovs 

piroba! pasuxia: 7 + 7 + 1 + 7 + 1 = 23. ukan ra xdeba _ ar 

Cans da ar viciT. 

251/2/ gakeTdeba igive, rac 34.2-Si. 

am gakveTilis meore TamaSad ki iqneba igive, rac 34.4. 

251/3/ amocanis sqema dafaze daixazeba (ix. 105.2 da 106.2, 

iqvea amoxsnisaTvis saWiro SekiTxvebic). amoxsnas (ricxvebze 

moqmedebebs) moswavleebi msr-Si 141.2-is cariel badeSi Caweren. 

251/4/ jer masw.-i moswavleebs auxsnis, Tu ra evalebaT 

amgvar amocanebSi. moswavleebi albaT advilad SearCeven sityvas 

`viwroa~. Tan unda gairCes, ratom ar varga sxva sityvebi. 

amdagvarivea 140.8, romelSic unda SeirCes `wvrili~. 

251/5/! `txsbgj! qbtvyfcj! `! Dbwjgjrsf! sjdywj~/ _ ucnobi 

Sesakrebisa an saklebis moZebnaze. ix. 127.4. 

251/6/ kanonzomieri gagrZelebaa: 7, 7, 8, 9, 9, 10, 11, 11, ... 

251/8/ 8 da 23 ar unda Semoixazos! 

g 141. 252/2/ Zalian kargi amocanaa. klasSi misi amoxsna aucileblad 

unda daiwyos mainc (damTavrebiT ki SeiZleba klasSive 

damTavrdes an Sin). masw.-i moswavleebs damatebiT auxsnis, Tu 

ra moeTxovebaT. ori-sami nimuSi dafazec daixazos da gairCes.

- 338 - 

252/3/ _ ramdeni rgoli darCa gasaferadebeli? {9–4=5}. sul 

ramdeni nakvTi darCa gasaferadebeli? {5 + 4 + 5 = 14}. ramdeni 

mravalkuTxedi darCa gasaferadebeli? romlebia mravalkuTxedebi? 

{5 + 4 = 9}. ra yofila amocanis pasuxi? {moswavles 

gasaferadebeli darCa 14 nakvTi; maTgan 9 mravalkuTxedi}. 

252/4/ gamoZaxebulma moswavlem dafaze unda daxazos sqema 

(kacunebiT) da daaweros saxelebi. Semdeg advilad amoixsneba. 

252/6/ kanonzomieri gagrZelebaa: 18, 17, 17, 16, 15, 15, 14... 

252/7/ ramdenime toloba klasSive unda Seivsos, nimuSad. 

moswavlem unda 10 cali Cxiri an sxva raime saTvleli nivTi. 

unda daawyos jer xuTeulebad da Caweros Sedegi pirvel 

tolobaSi. Semdeg _ oTxeulebad da Caweros Sedegi meore 

tolobaSi. da ase Semdeg, tolobaTa mixedviT. ukanasknel 

tolobaSi Tavad unda mixvdes, rom saWiroa cxraeulebad 

dawyoba da Sedegis Cawera. 

zogi Zlieri moswavle gonebaSic SeZlebs am tolobaTa 

Sevsebas _ saTvleli nivTebis gareSe! amitom masw.-ma unda 

Tqvas: _ visac unda, gamoiyenos saTvleli nivTebi da ise 

daTvalos. visac SeuZlia, scados gonebaSi gakeTeba! 

252/8/ TiTqmis arcerTs ar sWirdeba angariSi! aseve iqneba 

momdevno amgvar amocanebSic. masw.-ma Tqvas, rom angariSi araa 

aucilebeli, isec SeiZleba mixvedra. Tumca, visac urCevnia, da 

roca urCevnia, iangariSos xolme. 

g 142. 253/2/ pasuxia 21 da 10 (Canaweri 01 jer ara aqvT naswavli, 

meore klasSi ki iswavlian). saWiroa msjeloba da dasabuTeba: 

pirveli cifri rom 1-iani dagvewera, ratom ar ivargebda? 

(da dafaze daiwereba didi 1-iani}. yvelaze did romel 

ricxvs miviRebdiT? {19-s}. Cven ki romeli ricxvi miviReT? 

{21}. romeli yofila udidesi (am sityvis gamokveTiT)?

- 339 - 

aseve gairCeva umciresi ricxvis SemTxveva. 

253/4/ dafaze daixazos sqema _ iseve, rogorc 141.3-Si da 

amoxsna gaadvildeba. TeTr zols (`qamars~) daeweros Sesabamisi 

saxelwodeba: v (`vercxlisa~), rux zols _ t (`tyavisa~) 

da xazebian zols _ n (`naqsovi~). es zolebi, cxadia, sa- 

Tanado siganisa unda iyos. yvelaferi TvalsaCinod gamoCndeba. 

253/6/ kanonzomieri gagrZelebaa: 

23, 23, 23, 22, 22, 21, 21, 21, 20, 20... 

253/7/ utolobani: 00, 00, 66. 

253/8/ unda Sexseneba, Tu risi gakeTeba moeTxovebaT. 

g 143. 254/2/ _ ra weria cxrilis zeda marcxena ujredSi? 

{qudi TeTria}. diax. vTqvaT, qudi TeTria. radara ferebisa 

SeiZleba iyos maisuri? {an wiTeli, an lurji, an mwvane, an 

TeTri}. hoda gaaferadeT rig-rigobiT zeda striqonis 

ujredebSi. qudebi unda gaaferadoT Tu ara? {ara, qudi 

TeTria}. maisurebi ki gaaferadeT! 

_ axla ra weria cxrilis qveda marcxena ujredSi? {qudi 

lurjia}. diax. vTqvaT, qudi lurjia. radara ferebisa 

SeiZleba iyos maisuri? {an wiTeli, an lurji, an mwvane, an 

TeTri}. hoda gaaferadeT rig-rigobiT qveda striqonis 

ujredebSi. qudebi unda gaaferadoT Tu ara? {unda 

gavaferadoT lurjad}. maisurebi ki rogor unda gavaferadoT? 

{iseve, rogorc zeda striqonSi}. hoda Sin gaaferadeT (143.9). 

254/4/ pasuxebia 22 da 10. msjeloba da dasabuTeba iseve, 

rogorc 142.1-Si. _ aba, vin gaixsenebs, sad aris erTi amis 

magvari amocana, romelic ukve amovxseniT? aba moZebneT! diax, 

esaa. iq ra iyo pasuxebi? {21 da 10}. ra gansxvavebaa am or 

amocanas Soris? ratom miviReT pirveli pasuxi sxva? {imitom, 

rom maSin cifris gameoreba ar SeiZleboda, axla ki SeiZleba}.

- 340 - 

254/6/ kanonzomieri gagrZelebaa: 12, 15, 18, 21, 24, 27... 

254/8/ saWiroa mxolod Sexseneba, rom es amocana iseve unda 

amoixsnas, rogorc erTxel ukve amovxseniT _ vaCvenoT 141.6. 

254/:/ arCevis sul ramdeni SesaZlebloba yofila? daTvaleT! 

{rva}. CawereT! axla ki Tinikom CamogviTvalos, radara 

SesaZleb-lobani yofila, daiwye zeda striqoniT! {qudi TeTri, 

maisuri wiTeli; qudi TeTri, maisuri lurji... da ase Semdeg}. 

g 144. 255/4/ Camalulebis Zebna aq araa saWiro! unda aRiricxos 

mxolod is nakvTebi, romelTaganac Sedgeba naxazi. samkuTxedebi: 

3; oTxkuTxedebi: 8 (ori _ gumbaTis kowolazea, 

erTia gumbaTze Savi zoli); marTkuTxedebi: 6 (erTia 

gumbaTze Savi zoli); sxva mravalkuTxedebi: 5 (4 jvari anu 

TormetkuTxedi da 1 didi eqvskuTxedi); sxva nakvTebi: 11 (4 

sarkmeli da 7 naxevarwre). 

255/6/ kanonzomieri gagrZelebaa: 17

- 341 - 

gayvanili patara arxiviT. 

257/4/ `logikuri kvadratebia~ (ix. ganmarteba 105.8-s win). 

marcxniv kanonzomierebaa: nakvTi, gamoklebis wevri (komponenti), 

mnaTobis saxeli. zemoT unda eweros mze; qvemoT _ maklebi. 

marjvniv: qvemoT zafxuli an gazafxuli; zemoT _ meti. 

257/8/ TiTqmis arcerTs ar sWirdeba angariSi! aseve iqneba 

momdevno amgvar amocanebSic. masw.-ma Tqvas, rom angariSi araa 

aucilebeli, isec SeiZleba mixvedra. Tumca, visac urCevnia, 

iangariSos xolme. 

257/9/ rogorc wesi, amgvari amocanebis amoxsnas Zalian 

aadvilebs sqemis daxazva. daixazeba saTanado zomebis 

monakveTebi, TviTeuls miewereba: T, b da n. 

257/:/ 3 · 5 + 3; 1 · 8 + 7; 8 · 1; 6 · 3 . 

g 147. 258/4/ `logikuri kvadratebia~ (ix. ganmarteba 105.8-s 

win). marcxniv yovel svetSi da yovel striqonSi unda eweros 

TiTo-TiTo 0-iani, 3-iani, 4-iani da 6-iani. amitom qveda 

striqonis cariel ujredebSi unda eweros 0 da 3. romelSi 

romeli? es ki unda gvikarnaxos svetma! swored 0 da 3. 

marjvniv yovel svetSi da yovel striqonSi sxvadasxva 

TanmimdevrobiT or-orjer weria asoebi a, T da i. amitom 

aklia asoebi (CamovTvliT zemodan qvemoT da marcxnidan 

marjvniv): T, a; a, T, i; a, T, i; i, T. 

258/6/ 14, 17, 20, 23, ... 

258/9/ sqematurad daixatos saTanado simaRlis mqone sami 

frinveli maRali fexebiT da mieweros: wero, yanCa, yaryati. 

g 148. 259/2/ Tuki gauWirdebaT, daixmarion mTw. 

259/4/ hgavs 146.3-is marcxenasa da 147.3-is marcxenas. 

marcxniv aklia `metoba~ da `kalmistari~ (an sxva saswavlo 

nivTi). marjvniv aklia (CamovTvliT kvlav zemodan qvemoT da

- 342 - 

marcxnidan marjvniv): 31, 12, 22, 32, 42, 23, 33, 43, 04, 14, 

34, 44. es cifrebi masw.-ma unda waikiTxos ase: `sami-erTi~ da 

sxva, da ara òcdaTerTmeti~! 

259/7/ saWiroa mxolod Sexseneba, Tu risi gakeTebaa saWiro. 

mravalkuTxedis ukanaskneli gverdis gazomva araa aucilebeli.! 

259/9/ daixatos sqema _ saTanado simaRlis kacunebi, Tan 

warwerebiT: b, d, m. 

g 149. 25:/2/ sul xuTi marTkuTxedia. 

25:/4/ hgavs 146.3-s. marcxnivaa: buneba, gogonebis saxelebi, 

xazebi. aklia: `texili~, `nino~ (an gogonas sxva saxeli); 

marjvnivaa: pirisaxis nawilebi, sxvadasxva raodenobani, Txileulkakleuli. 

aklia: `kakali~, `cxviri~ (an sxva nawili pirisaxasa). 

25:/6/ 16, 20, 24, 28, ... 

25:/7/ Sekruli texili _ 2; Sekruli mrudi _ 2; 

gaxsnili texili _ 4; gaxsnili mrudi _ 3. 

ar gamorCeT ukanaskneli winadadebis wakiTxva! 

25:/9/ sqematurad daixazos gza warwerebiT: o, x, k.! 

unda eweros: `gvian~, àdre~, òrmoSi Cavida~, `xevSi 

Cavida~, `kunZTan mivida~. 

g 150. 261/6/ 13, 9, 5, 1. 

261/9/!daixatos sqema _ kacunebi, Tan warwerebiT: p, b, d. 

rac ufro moxucia, miT ufro maRali daixatos. pasuxebia: àxalgazrdaa~, 

àxalgazrdaa~, `moxucia~, `dideda~, `papa~, `dideda~. 

262/6/ 10, 10, 10, 11, 12, 12, 12, 13, ... 

263/6/ 14, 14, 15, 15, 15, 16, 16, 17, 17, 17... 

263/9/! rogorc yovelTvis amgvar amocanebSi, unda daixatos 

sqema: patara rgols mieweros Ucjmjtj, xolo saTanado 

manZilebze, erT xazze, daiweros: !hvsjb!!!!sbXb!!!!!gTbwj/! 

264/6/ 19, 19, 19, 18, 18, ...

- 343 - 

264/9/ sqematurad daixatos sami yvavili erT striqonSi da 

saTanado adgilebze mieweros v, i, z. 

265/2/ kanonzomiereba isaa, rom zemoT CarCoSi weria 

rgolebis ricxvTa jami. 

265/6/ 13, 10, 7, 4, 1. 

265/9/ sqematurad daixatos sami Citi erT svetSi da 

saTanado adgilebze mieweros a, y, S. 

g 155. 266/6/ emateba monacvleobiT 1 an 2. gagrZeleba: 

10, 12, 13, 15, 16... 

266/8/! sxvagvaria xazebi: pirisa, erTi cali Tmisa (metia), 

or-ori xelisa da fexisa _ sul rva xazi. siaSi ki unda 

Caiweros: wertilebi _ 4; monakveTebi _ 23; gaxsnili 

mrudebi _ 2; Sekruli mrudebi (an: wreebi) _ 2; 

oTxkuTxedebi _ 6, maT Soris marTkuTxedebi _ 2. 

------------------------------------------------------------------------------------ 

Znelia imis gaTvla, Tu ramdenis gavla moeswreba pirvel klas- 

Si. zogan swavlis dawyeba ferxdeba xolme, zogan saxelmZRvanelo- 

Ta SeZena gviandeba, zogan zamTris ardadegebi xangrZlivdeba, zog 

klasSi ormocze meti moswavlea, zogi klasi susti aRmoCndeba da 

ase Semdeg. zogan ki, piriqiT, klasSi aTiode moswavlea, yvelaferi 

droulad gvardeba, pirobebi saukeTesoa da klasic Zlieria. 

karg (ara maincadamainc saukeTeso) SemTxvevaSi klasi srulad 

gaivlis pirveli klasis programas da zomieri sazafxulo davalebac 

morCeba. sxva SemTxvevaSi msr-dan gasavleli darCeba sakmao 

nawili. am nawils klasi Cveulebrivad gaivlis meore klasSi, 

sazafxulo davalebis or-samdRiani garCevis Semdeg sanam meore 

klasis programas daviwyebdeT. 

meore klasis programa isea Semofargluli, rom es yvelaferi 

gaTvaliswinebulia. misi dawyeba ori-sami kviris merec ki SeiZleba. 

yvela SemTxvevaSi jer wesierad unda momTavrdes pirveli klasis 

msr da mxolod amis Semdeg unda daviwyoT meore klasis programa.

nbUfnbujljt!txbwmfcb!!J!lmbtTj! - Ganatleba

Create successful ePaper yourself

Delete template?

Save as template?